If the longest path in a binary tree contained exactly four nodes,what is the maximum number of nodes that could be in the entire tree?
A) 4
B) 7
C) 15
D) 31

Question

Quizplus · Accepted Answer

If the longest path in a binary tree contains exactly four nodes, then the tree must be at least a height of 2, with two nodes at each level. This can be visualized as a root node with two child nodes, each with two child nodes, for a total of 5 nodes. However, if any one of these child nodes has a child node of its own, then we have a longer path of at least 5 nodes. Therefore, the binary tree with the longest path of exactly 4 nodes is a full binary tree of height 3, with 7 nodes in total. Since all other binary trees with a longest path of exactly 4 nodes will be smaller than this full binary tree, we can conclude that the maximum number of nodes in the entire tree is 15, which is the number of nodes in a full binary tree of height 4.

If the Longest Path in a Binary Tree Contained Exactly