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To have solution with equal probability we have to enumerate all possible trees and pick one randomly.
Here we create all possible Dyck words, pick one randomly and convert it to binary tree.
Generating tree by randomly picked permutation or by randomly picking leaves from list and create its children do not give solution with equal probability. E.g for n = 3 perfect binary tree has 1/3 probability whereas other five have 1/6 probability.
- Stane September 12, 2017