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(a) Show that in a binary tree, there must be at least two leaf nodes that are siblings. [If a binary tree is defined as a tree formed by a certain process, this proposition is evident; the last two children formed will be siblings and leaf nodes. Here is another proof. Take a node a greatest distance from the root node. It and its sibling must be leaf nodes. Why?]
(b) Show that a binary tree with m leaf nodes has 2m −1 nodes, total. [Use (a) and go by induction on m.]