Right Path in a Leftist Tree is Short
If the right path has length
at least r, the tree has at
Basis: r = 1. Tree has at least one node: 21 - 1 = 1
Inductive step: assume true for r’ < r. The right subtree has a right path of at least r - 1 nodes, so it has at least 2r - 1 - 1 nodes. The left subtree must also have a right path of at least r - 1 (otherwise, there is a null path of r - 3, less than the right subtree). Again, the left has 2r - 1 - 1 nodes. All told then, there are at least:
2r - 1 - 1 + 2r - 1 - 1 + 1 = 2r - 1
So, a leftist tree with at least n nodes has a right path of at most log n nodes