The top eigenvalue of uniformly random trees

Let Tₙ be a uniformly random tree with vertex set n=\1, , n\, let ₓₙ be the largest vertex degree in Tₙ, and let ₁ (Tₙ) be the largest eigenvalue of its adjacency matrix. We prove that |₁ (Tₙ) - ₓ䂸| 0 in expectation as n, and additionally prove probability tail bounds for |₁ (Tₙ) - ₓ䂸|. The proof is based on the trace method and thus on counting closed walks in a random tree. To this end, we develop novel combinatorial tools for encoding walks in trees that we expect will find other applications. In order to apply these tools, we show that uniformly random trees -- after appropriate "surgery" -- satisfy, with high probability, the properties required for the combinatorial bounds to be effective.