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Tree Traversals (Inorder, Preorder and Postorder) - GeeksforGeeksPosted here for convenience.Tree Traversals (Inorder, Preorder and Postorder)Unlike linear data structures (Array, Linked List, Queues, Stacks, etc) which have only one logical way to traverse them, trees can be traversed in different ways. Following are the generally used ways for traversing trees.Example TreeDepth First Traversals:(a) Inorder (Left, Root, Right) : 4 2 5 1 3(b) Preorder (Root, Left, Right) : 1 2 4 5 3(c) Postorder (Left, Right, Root) : 4 5 2 3 1Breadth First or Level Order Traversal : 1 2 3 4 5Please see this post for Breadth First Traversal.Inorder Traversal (Practice):Algorithm Inorder(tree)  1. Traverse the left subtree, i.e., call Inorder(left-subtree)  2. Visit the root.  3. Traverse the right subtree, i.e., call Inorder(right-subtree) Uses of InorderIn case of binary search trees (BST), Inorder traversal gives nodes in non-decreasing order. To get nodes of BST in non-increasing order, a variation of Inorder traversal where Inorder traversal s reversed can be used.Example: Inorder traversal for the above-given figure is 4 2 5 1 3.Preorder Traversal (Practice):Algorithm Preorder(tree)  1. Visit the root.  2. Traverse the left subtree, i.e., call Preorder(left-subtree)  3. Traverse the right subtree, i.e., call Preorder(right-subtree)  Uses of PreorderPreorder traversal is used to create a copy of the tree. Preorder traversal is also used to get prefix expression on of an expression tree. Please seehttp://en.wikipedia.org/wiki/Polish_notation to know why prefix expressions are useful.Example: Preorder traversal for the above given figure is 1 2 4 5 3.Postorder Traversal (Practice):Algorithm Postorder(tree)  1. Traverse the left subtree, i.e., call Postorder(left-subtree)  2. Traverse the right subtree, i.e., call Postorder(right-subtree)  3. Visit the root. Uses of PostorderPostorder traversal is used to delete the tree. Please see the question for deletion of tree for details. Postorder traversal is also useful to get the postfix expression of an expression tree. Please seehttp://en.wikipedia.org/wiki/Reverse_Polish_notation to for the usage of postfix expression.Example: Postorder traversal for the above given figure is 4 5 2 3 1.C++// C program for different tree traversals #include <iostream> using namespace std; /* A binary tree node has data, pointer to left child and a pointer to right child */struct Node { int data; struct Node* left, *right; Node(int data) { this->data = data; left = right = NULL; } }; /* Given a binary tree, print its nodes according to the "bottom-up" postorder traversal. */void printPostorder(struct Node* node) { if (node == NULL) return; // first recur on left subtree printPostorder(node->left); // then recur on right subtree printPostorder(node->right); // now deal with the node cout << node->data << " "; } /* Given a binary tree, print its nodes in inorder*/void printInorder(struct Node* node) { if (node == NULL) return; /* first recur on left child */ printInorder(node->left); /* then print the data of node */ cout << node->data << " "; /* now recur on right child */ printInorder(node->right); } /* Given a binary tree, print its nodes in preorder*/void printPreorder(struct Node* node) { if (node == NULL) return; /* first print data of node */ cout << node->data << " "; /* then recur on left sutree */ printPreorder(node->left); /* now recur on right subtree */ printPreorder(node->right); } /* Driver program to test above functions*/int main() { struct Node *root = new Node(1); root->left = new Node(2); root->right = new Node(3); root->left->left = new Node(4); root->left->right = new Node(5); cout << "\nPreorder traversal of binary tree is \n"; printPreorder(root); cout << "\nInorder traversal of binary tree is \n"; printInorder(root); cout << "\nPostorder traversal of binary tree is \n"; printPostorder(root); return 0; } CPythonJavaC#Output:Preorder traversal of binary tree is 1 2 4 5 3  Inorder traversal of binary tree is 4 2 5 1 3  Postorder traversal of binary tree is 4 5 2 3 1 One more example:Time Complexity: O(n)Let us see different corner cases.Complexity function T(n) — for all problem where tree traversal is involved — can be defined as:T(n) = T(k) + T(n – k – 1) + cWhere k is the number of nodes on one side of root and n-k-1 on the other side.Let’s do an analysis of boundary conditionsCase 1: Skewed tree (One of the subtrees is empty and other subtree is non-empty )k is 0 in this case.T(n) = T(0) + T(n-1) + cT(n) = 2T(0) + T(n-2) + 2cT(n) = 3T(0) + T(n-3) + 3cT(n) = 4T(0) + T(n-4) + 4c…………………………………………………………………………………….T(n) = (n-1)T(0) + T(1) + (n-1)cT(n) = nT(0) + (n)cValue of T(0) will be some constant say d. (traversing a empty tree will take some constants time)T(n) = n(c+d)T(n) = Θ(n) (Theta of n)Case 2: Both left and right subtrees have equal number of nodes.T(n) = 2T(|_n/2_|) + cThis recursive function is in the standard form (T(n) = aT(n/b) + (-)(n) ) for master method http://en.wikipedia.org/wiki/Master_theorem. If we solve it by master method we get (-)(n)Auxiliary Space : If we don’t consider size of stack for function calls then O(1) otherwise O(n).Recommended Posts:Check if given Preorder, Inorder and Postorder traversals are of same treePreorder from Inorder and Postorder traversalsPrint Postorder traversal from given Inorder and Preorder traversalsConstruct Tree from given Inorder and Preorder traversalsConstruct Full Binary Tree from given preorder and postorder traversalsConstruct a Binary Tree from Postorder and InorderConstruct a tree from Inorder and Level order traversals | Set 1Construct a tree from Inorder and Level order traversals | Set 2Construct Full Binary Tree using its Preorder traversal and Preorder traversal of its mirror treeFind postorder traversal of BST from preorder traversalPostorder successor of a Node in Binary TreeConstruct a Binary Search Tree from given postorderFind n-th node in Postorder traversal of a Binary TreePostorder predecessor of a Node in Binary Search TreePostorder traversal of Binary Tree without recursion and without stackImproved By : danielbritten, andrew1234Article Tags : ArticlesTreeInorder TraversalPostOrder TraversalPreorder TraversalSnapdealtree-traversalTreesPractice Tags :SnapdealTreethumb_up27To-doDone1.5Based on 371 vote(s)Feedback/ Suggest ImprovementAdd NotesImprove ArticlePlease write to us at [email protected] to report any issue with the above content.Post navigationPreviousfirst_pageNextlast_pageWrite a program to Calculate Size of a tree | RecursionWriting code in comment? 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How long is Covid active on packages (cardboard), door knobs (steel, copper), plastic, and in the air?[Update 1/19/21]: A newer Oct 2020 study shows up to 14 days on cloth under indoor conditions (no sunlight). 28+ days on other surfaces.At 20 °C, infectious SARS-CoV-2 virus was still detectable after 28 days post inoculation, for all non-porous surfaces tested (glass, polymer note, stainless steel, vinyl and paper notes). The recovery of SARS-CoV-2 on porous material (cotton cloth) was reduced compared with most non-porous surfaces, with no infectious virus recovered past day 14 post inoculation.The effect of temperature on persistence of SARS-CoV-2 on common surfaces (Virology Journal, 10/7/20)Covid virus ‘survives for 28 days’ in lab conditions (BBC, 10/11/20)Back in Mar 2020, there were two major studies: Hong Kong & US.(1) Hong Kong University: 3/15/20 preprint (4/2/20 published in Lancet)Studied pipetted dropletsPaper & Tissue: 30 min+, undetectable at 3 hrsWood, Cloth: 1 day+, undetectable at 2 daysGlass, Banknote: 2 days+, undetectable at 4 daysSteel, Plastic, Surgical Mask (inner layer): 4 days+, undetectable at 7 daysSurgical Mask (outer layer): 7 days+5 min hand soap is not long enough to inactivate virus.20 sec of hand-washing might be enough to flush virus down the sink.- Stability of SARS-CoV-2 in different environmental conditions (Chin et al., medRxiv preprint, 3/15/20)- Stability of SARS-CoV-2 in different environmental conditions (Chin et al., Lancet, 4/2/20). Supplemental tables https://www.thelancet.com/cms/10.1016/S2666-5247(20)30003-3/attachment/cb3624c2-996e-4202-a337-5cf60df1021e/mmc1.pdf (link in the paper doesn’t seem to work properly on my laptop).(2) US: Princeton, UCLA, NIH (National Institutes of Health): 3/9/20 preprint (3/17/20 published in NEJM)Studied aerosols.Air: 3 hoursCardboard: 1 day, 2 days to be undetectableSteel: 2–3 days, 4 days to be undetectablePlastic: 3 days, 4 days to be undetectableAt room temperature. Scientists store viruses in freezers for years.Note: No data yet on fabric (clothing), different temperatures or humidities.- How long can the novel coronavirus survive on surfaces and in the air? (Economist diagram, 3/19/20)How to clean packages:Hello I'm not trying to panic but my question is I ordered a package from China before the outbreak is this package safe for me to open once it gets here I know a lot of people are wondering the same thing? in Coronavirus WatchIf you don’t want a lot of details, you can skip the rest of this answer.Stay safe.Medium-length answer:(1) Hong Kong University: 3/15/20 preprint (4/2/20 published in Lancet)Studied pipetted droplets. This research group also looked at virus lifetimes at different temperatures.It is poorly written. Unclear what temperature these surfaces were studied at.Paper & Tissue: 30 min+, undetectable at 3 hrsWood, Cloth: 1 day+, undetectable at 2 daysGlass, Banknote: 2 days+, undetectable at 4 daysSteel, Plastic, Surgical Mask (inner layer): 4 days+, undetectable at 7 daysSurgical Mask (outer layer): 7 days+5 min hand soap is not long enough to inactivate virus.20 sec of hand-washing might be enough to flush virus down the sink.- Stability of SARS-CoV-2 in different environmental conditions (Chin et al, medRxiv preprint, 3/15/20)- Stability of SARS-CoV-2 in different environmental conditions (Chin et al., Lancet, 4/2/20). Supplemental tables https://www.thelancet.com/cms/10.1016/S2666-5247(20)30003-3/attachment/cb3624c2-996e-4202-a337-5cf60df1021e/mmc1.pdf (link in the paper doesn’t seem to work properly on my laptop).(2) US: Princeton, UCLA, NIH (National Institutes of Health): 3/9/20 preprint (3/17/20 published in NEJM)Update 3/21/20: Previous half-life numbers from a 3/9/20 preprint were wrong. They have been revised (dropped by 50–70%) in the final 3/17/20 version (after peer-review).From bioXriv preprint by Princeton, UCLA, and National Institutes of Health (NIH) (3/9/20) and final NEJM correspondence (3/17/20)Old pre-print Covid half-lives were TOO HIGH:Aerosol 2.74 hrs (1.65–7.24 hrs)Copper 2.4 (2.4–5.11)Cardboard 8.45 (5.95–12.4)Steel 13.1 (10.5–16.1)Plastic 15.9 (13–19.2)Final NEJM version half-lives:Boldface numbers are visual estimates from small graphs.Regular numbers are from text in the paper.Copper 0.8 hrs (0.43–1.3 hrs)Aerosol 1.1–1.2 hrs (0.64–2.64 hrs)Cardboard 3.5 hrs (2–5 hrs)Steel 5.6 hrs (4.5–7 hrs)Plastic 6.8 hrs (5.5–8.5 hrs)- Aerosol and Surface Stability of SARS-CoV-2 as Compared with SARS-CoV-1 | NEJM (NEJM final correspondence, 3/17/20)——Long version (more than you ever wanted to know):Twitter comments from Dr. Angela Rasmussen about the impact of this paper.“…suggesting that HCoV-19 doesn't last more than a day on cardboard. People who are concerned about receiving packages should take note. People who handle packages/delivery people should take extra hand hygiene/protection precautions.”- Dr. Angela Rasmussen on Twitter (3/11/20)“The key takeaway is that #SARSCoV2 #HCoV19 is very similar to SARS classic in how long the virus can remain infectious in all these circumstances. Both retain infectivity longest on plastic and stainless steel, although this is reduced substantially after 48-72 hours.Furthermore, both viruses also showed a 3-log decrease in infectious virus on stainless steel and plastic surfaces after 48-72 hours. This suggests that, while viruses can stay on some surfaces for days, their infectious titer is greatly reduced (1000-fold).”- Dr. Angela Rasmussen on Twitter (3/11/20)“This data suggests that aerosolized virus half-life is measured in hours, and aerosols generated by these medical procedures will not persist for days, for example in the hospital procedure rooms.Both SARS-2 and SARS classic viruses remain infectious after 3 hours as experimentally-generated aerosols. However, there is a 1-log reduction in virus titer, meaning that the number of infectious virus units decreased by a factor of 10 after 3 hours.It's also important to note that neither SARS-CoV classic nor SARS-CoV-2/HCoV19/COVID-19 normally form aerosols. These are only generated under very specific circumstances during certain hospital procedures (bronchoscopy, intubation).”- Dr. Angela Rasmussen on Twitter (3/11/20)Diagrams and text excerpts from preprint:Note: The first four diagrams are ok.The table and aerosol chart are incorrect. Possibly, reviewers pointed out to the authors that calculations were off.Note: This TEXT IS OUT-OF-DATE.“We found that viable virus could be detected in aerosols up to 3 hours post aerosolization, up to 4 hours on copper, up to 24 hours on cardboard and up to 2-3 days on plastic and stainless steel.HCoV-19 and SARS-CoV-1 exhibited similar half-lives in aerosols, with median estimates around 2.7 hours.Both viruses show relatively long viability on stainless steel and polypropylene compared to copper or cardboard: the median half-life estimate for HCoV-19 is around 13 hours on steel and around 16 hours on polypropylene.Our results indicate that aerosol and fomite transmission of HCoV-19 is plausible, as the virus can remain viable in aerosols for multiple hours and on surfaces up to days.We found that the half-life of HCoV-19 on cardboard is longer than the half-life of SARS-CoV-1. It should be noted that individual replicate data were noticeably noisier for this surface than the other 5 surfaces tested (Figures S1–S5), so we advise caution in interpreting this result.”- Aerosol and surface stability of HCoV-19 (SARS-CoV-2) compared to SARS-CoV-1 https://www.medrxiv.org/content/10.1101/2020.03.09.20033217v1.full.pdf (3/9/20)Newer information (3/17/20) with updated half-life numbers:“The half-lives of SARS-CoV-2 and SARS-CoV-1 were similar in aerosols, with median estimates of approximately 1.1 to 1.2 hours and 95% credible intervals of 0.64 to 2.64 for SARS-CoV-2 and 0.78 to 2.43 for SARS-CoV-1.The half-lives of the two viruses were also similar on copper.On cardboard, the half-life of SARS-CoV-2 was longer than that of SARS-CoV-1.The longest viability of both viruses was on stainless steel and plastic; the estimated median half-life of SARS-CoV-2 was approximately 5.6 hours on stainless steel and 6.8 hours on plastic.”- Aerosol and Surface Stability of SARS-CoV-2 as Compared with SARS-CoV-1 | NEJM (NEJM final correspondence, 3/17/20)Older info on SARS-CoV, MERS-CoV, and “common cold” viruses.“Coronaviruses like the Wuhan virus can travel only about six feet from the infected person. It’s unknown how long they live on surfaces.Some other viruses, like measles, can travel up to 100 feet…”- How Bad Will the Coronavirus Outbreak Get? Here Are 6 Key Factors (New York Times, 1/31/20)“Both influenza virus and SARS-CoV surrogates have been shown to survive for extended periods on N95 respirator material. This survival represents a barrier to the reuse of N95 respirators. One approach is to disinfect the N95 respirators. Several candidate technologies have been evaluated for the disinfection of N95 respirators; UV light, hydrogen peroxide vapour, and ethylene oxide show most promise.We reviewed the capacity of viruses with pandemic potential, influenza SARS-CoV and MERS-CoV, to survive on dry surfaces. The experimental methods used to test survival are important, but it seems that surface survival of SARS/MERS-CoV is greater than that of influenza virus.”- Transmission of SARS and MERS coronaviruses and influenza virus in healthcare settings: the possible role of dry surface contamination. (2015)“The human coronavirus associated with the common cold was reported to remain viable only for 3 hours on environmental surfaces after drying, although it remains viable for many days in liquid suspensionIn the present study, we have demonstrated that SARS CoV can survive at least two weeks after drying at temperature and humidity conditions found in an air-conditioned environment. The virus is stable for 3 weeks at room temperature in a liquid environment but it is easily killed by heat at 56°C for 15 minutes. This indicates that SARS CoV is a stable virus that may potentially be transmitted by indirect contact or fomites.It may also explain why Singapore, which is also in tropical area, had most of its SARS outbreaks in hospitals (air-conditioned environment).Interestingly, during the outbreak of SARS in Guangzhou, clinicians kept the windows of patient rooms open and well ventilated and these may well have reduced virus survival and this reduced nosocomial transmission.SARS CoV can retain its infectivity up to 2 weeks at low temperature and low humidity environment, which might facilitate the virus transmission in community as in Hong Kong which locates in subtropical area.“- The Effects of Temperature and Relative Humidity on the Viability of the SARS Coronavirus (2011)“SARS-CoV has been reported to survive for 36 h on stainless steel…At ambient ATs (around 20°C), coronaviruses can survive for 2 days while losing only 1 to 2 log10 infectivity, depending on the RH.Nasopharyngeal aspirates from infected individuals with SARS can have viral loads ranging from 105 to 108 genome templates/ml, suggesting that respiratory secretions from SARS patients may contain infectious virus.If deposited on surfaces in these types of secretions, coronaviruses could potentially survive on surfaces in health care environments for days.Evidence of SARS-CoV nucleic acids on surfaces and inanimate objects in hospitals has been reported. However, there are no data on the occurrence of infectious SARS-CoV on these surfaces.The dose-response relationship and minimal infectious doses for infection of humans by SARS-CoV and other coronaviruses have also not been defined.”- Effects of Air Temperature and Relative Humidity on Coronavirus Survival on Surfaces (NIH, 2010)“Laboratory studies have shown that the SARS (severe acute respiratory syndrome) coronavirus can survive up to 2 days on plastic surfaces and at least that long in human feces, the World Health Organization (WHO) has announced.The studies, done by WHO network labs in Hong Kong, Japan, and Germany, also show that the virus can survive in urine for at least 24 hours, the WHO reported. Virus from the feces of patients suffering diarrhea, which is less acidic than normal stool, was found to survive for 4 days."However, the dose of virus needed to cause infection remains unknown," the WHO statement said.”- SARS virus can last 2 days on surfaces and in feces (WHO, 2003)“Many different types of viruses can cause colds. The viruses can sometimes survive on indoor surfaces for more than 7 days. In general, viruses survive for longer on non-porous (water resistant) surfaces, such as stainless steel and plastics, than porous surfaces, such as fabrics and tissues. Although cold viruses have been shown to survive on surfaces for several days, their ability to cause an infection reduces rapidly and they don't often survive longer than 24 hours.Most viruses which cause colds only survive on hands for a short amount of time. Some only last for a few minutes but 40% of rhinoviruses, a common cold-causing virus, are still infectious on hands after an hour.Respiratory syncytial virus (RSV), another cold-like virus that can cause serious illness in children, can survive on worktops and door handles for up to 6 hours, on clothing, and tissues for 30 to 45 minutes and on skin for up to 20 minutes.”- How long do bacteria and viruses live outside the body? (UK NHS, National Health Service)“Surgical masks are highly effective against large airborne droplets.These traditional masks are less effective with small droplets, as they can travel farther and in unpredictable paths affected by wind and other gusts. The droplets can be inhaled around the sides of the masks.”- Blisse Edgerton's answer to What should you know about the new strain of coronavirus in China?Persistence of coronaviruses on inanimate surfaces and their inactivation with biocidal agents (2/12/20) is an analysis of 22 previous SARS/MERS/virus studies.It’s the source of “survive up to 9 days on uninfected surface, study finds”.See also Mitchell Tsai's answer to How serious is the 2019–20 Coronavirus?