On the maximum of a simple random walk

Let $S_0=0$, $S_n=\xi_1+\xi_2+\dots+\xi_n$, $n\ge 1$, be the simple random walk generated by a sequence of independent random variables $\xi_i $, $i=1,2,\dots$, such that $\mathbf{P}\{\xi_i=1\}=1-\mathbf{P}\{\xi_i=-1\}=\frac12$, and let $T$ be the moment of the first return of $S_n$ to the state 0. We find an asymptotic representation for the probability $\mathbf{P}\{\max_{0<k<T}|S_k|>n|T=2N\}$ which is exact (in order), assuming that $n^2 N^{-1}\to\infty$, and $nN^{-1}\le a<1$. The results obtained are used to study the asymptotics of moderate and large deviations of the height of a planted plane tree with $N$ vertices.