When elements are given in a sequence, Always consider the first element as the root node. Presumably each node contains information about one of the 50 states. This is an example of a binary tree with nine nodes. Any binary tree can have at most 2d nodes at depth d. (Easy proof by induction) Also, in the last level, nodes should be attached starting from the left-most position. In this example, the states are not arranged in any particular order, except insofar as I need to illustrate the different special kinds of nodes and connections in a binary tree. What is Complete Binary Tree? Nearly Complete Binary Trees and Heaps DEFINITIONS: i) The depth of a node p in a binary tree is the length (number of edges) of the path from the root to p. ii) The height (or depth) of a binary tree is the maxi-mum depth of any node, or −1 if the tree is empty. Binary Search Tree Construction- Let us understand the construction of a binary search tree using the following example- Example- Construct a Binary Search Tree (BST) for the following sequence of numbers-50, 70, 60, 20, 90, 10, 40, 100 . Almost complete Binary Tree A binary tree of depth d is an almost-binary tree if any node n at level d-1 has two children and for any node n in the tree with a right or left child at level l, the node must have a left child(if it has a right child) and all the nodes on the left side must have two children. As shown in figure 2, a complete binary tree is a binary tree in which every level of the tree is completely filled except the last level. In a complete binary tree every level, except possibly the last, is completely filled, and all nodes in the last level are as far left as possible. A binary heap is an almost complete binary tree, which is a binary tree that is completely filled, with the possible exception of the bottom level, which is filled left to right. Red-Black trees maintain O(Log n) height by making sure that the number of Black nodes on every root to leaf paths is the same and there are no adjacent red nodes. Practical example of Complete Binary Tree is Binary Heap. * A binary tree is a tree where each node has 0, 1 or 2 children * A heap is a tree with the heap property: the value of every node is larger than or equal the value of any one of its descendants. No! ... (Log n) height by making sure that the difference between the heights of the left and right subtrees is almost 1.

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