How to show that the spectral radius of a binary tree approaches exp(1) as the N tends to infinity?

Question

How can I prove mathematically that the spectral radius of a binary tree approaches e as the number of nodes tends to infinity? I mean it is true that the increase in nodes number is exponential but so far I have computed the spectral radius by finding the largest nontrivial eigenvalue of det [A(G)- lambda*I]=0. How can I prove that if N tends to infinity, the spectral radius is exp(1)??