The nodes are either a part of left sub tree or the right sub tree, so we don’t have to traverse all the nodes, this means the complexity is lesser than n, in the average case, assuming the nodes are spread evenly, the time complexity becomes O(logn).

A full binary tree is a tree in which each node has exactly 0 or 2 children.

The number of edges from the node to the deepest leaf is called height of the tree.

Number of Leaf nodes in full binary tree is equal to 1 + Number of Internal Nodes i.e L = I + 1

In a binary tree, there are atmost 2k nodes in level k and 2k-1 total number of nodes. Number of levels is at least ceil(log(N+1)).

The number of edges from the root to the node is called depth of the tree.

Relation between number of internal nodes(I) and nodes(N) is N = 2*I+1.

The relation between number of nodes(N) and leaves(L) is N=2*L-1.

A binary tree, which is completely filled, with the possible exception of the bottom level, which is filled from left to right is called complete binary tree. A Tree in which each node has exactly zero or two children is called full binary tree. A Tree in which the degree of each node is 2 except leaf nodes is called perfect binary tree.

Undo/Redo operations in a notepad is an application of stack. Hierarchical structure, Faster search, Router algorithms are advantages of trees.