  Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_n(\beta)$
associated to a supercritical branching random walk with mean number of
offspring $m$. We prove a functional central limit theorem stating that as
$n\to\infty$ the process $$ D_n(u):= m^{\frac 12 n}
\left(W_{\infty}\left(\frac{u}{\sqrt n}\right) - W_{n}\left(\frac{u}{\sqrt
n}\right) \right) $$ converges weakly, on a suitable space of analytic
functions, to a Gaussian random analytic function with random variance. Using
this result we prove central limit theorems for the total path length of random
trees. In the setting of binary search trees, we recover a recent result of R.
Neininger [Refined Quicksort Asymptotics, Rand. Struct. and Alg., to appear],
but we also prove a similar theorem for uniform random recursive trees.
Moreover, we replace weak convergence in Neininger's theorem by the almost sure
weak (a.s.w.) convergence of probability transition kernels. In the case of
binary search trees, our result states that $$ L\left\{\sqrt{\frac{n}{2\log n}}
\left(EPL_{\infty} - \frac{EPL_n-2n\log n}{n}\right)\Bigg | G_{n}\right\} \to
\{\omega\mapsto N_{0,1}\}, \quad \text{a.s.w.},$$ where $EPL_n$ is the external
path length of a binary search tree $X_n$ with $n$ vertices, $EPL_{\infty}$ is
the limit of the R\'egnier martingale, and $L(\,\cdot\, |G_n)$ denotes the
conditional distribution w.r.t. the $\sigma$-algebra $G_n$ generated by
$X_1,\ldots,X_n$. A.s.w. convergence is stronger than weak and even stable
convergence. We prove several basic properties of the a.s.w. convergence and
study a number of further examples in which the a.s.w. convergence appears
naturally. These include the classical central limit theorem for Galton-Watson
processes and the P\'olya urn.
