A tree is a widely used abstract data type that represents a hierarchical structure with a set of connected nodes. Each node in the tree can be connected to many children, but must be connected to exactly one parent, except for the root node, which has no parent.
A tree is an undirected and connected acyclic graph. There are no cycles or loops. Each node can be like the root node of its own subtree, making [recursion](recursion.md) a useful technique for tree traversal.
For the purpose of interviews, you will usually be asked on binary trees as opposed to ternary (3 children) or N-ary (N children) trees. In this page,we will cover binary trees and binary search trees, which is a special case of binary trees.
Trees are commonly used to represent hierarchical data, e.g. file systems, JSON, and HTML documents. Do check out the section on [Trie](trie.md), which is an advanced tree used for efficiently storing and searching strings.
- [A Brief Guide to Binary Search Trees](https://www.youtube.com/watch?v=0woI8l0ZWmA) ([slides](https://samuelalbanie.com/files/digest-slides/2022-10-brief-guide-to-binary-search-trees.pdf)), Samuel Albanie, University of Cambridge
- [A Brief Guide to Red-Black Trees](https://www.youtube.com/watch?v=t-oiZnplv7g) ([slides](https://samuelalbanie.com/files/digest-slides/2022-12-brief-guide-to-red-black-trees.pdf)), Samuel Albanie, University of Cambridge
- [A Brief Guide to B-trees](https://www.youtube.com/watch?app=desktop&v=7MqaHGWRS3E) ([slides](https://samuelalbanie.com/files/digest-slides/2022-12-brief-guide-to-b-trees.pdf)), Samuel Albanie, University of Cambridge
- Complete binary tree - A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes in the last level are as far left as possible.
- Balanced binary tree - A binary tree structure in which the left and right subtrees of every node differ in height by no more than 1.
Space complexity of traversing balanced trees is O(h) where h is the height of the tree, while traversing very skewed trees (which is essentially a linked list) will be O(n).
You should be very familiar with writing pre-order, in-order, and post-order traversal recursively. As an extension, challenge yourself by writing them iteratively. Sometimes interviewers ask candidates for the iterative approach, especially if the candidate finishes writing the recursive approach too quickly.
Recursion is the most common approach for traversing trees. When you notice that the subtree problem can be used to solve the entire problem, try using recursion.
When using recursion, always remember to check for the base case, usually where the node is `null`.
Sometimes it is possible that your recursive function needs to return two values.
### Traversing by level
When you are asked to traverse a tree by level, use breadth-first search.
### Summation of nodes
If the question involves summation of nodes along the way, be sure to check whether nodes can be negative.
- [Binary Tree Maximum Path Sum](https://leetcode.com/problems/binary-tree-maximum-path-sum/)
- [Binary Tree Level Order Traversal](https://leetcode.com/problems/binary-tree-level-order-traversal/)
- [Lowest Common Ancestor of a Binary Tree](https://leetcode.com/problems/lowest-common-ancestor-of-a-binary-tree/)
- [Binary Tree Right Side View](https://leetcode.com/problems/binary-tree-right-side-view/)
- [Subtree of Another Tree](https://leetcode.com/problems/subtree-of-another-tree/)
- [Construct Binary Tree from Preorder and Inorder Traversal](https://leetcode.com/problems/construct-binary-tree-from-preorder-and-inorder-traversal/)
- [Serialize and Deserialize Binary Tree](https://leetcode.com/problems/serialize-and-deserialize-binary-tree/)
