Let $B$ be a standard one-dimensional Brownian motion started at 0. Let $L_{t,v}(|B|)$ be the occupation density of $|B|$ at level $v$ up to time $t$. The distribution of the process of local times $(L_{t,v}(|B|), v \ge 0 )$ conditionally given $B_t = 0$ and $L_{t,0} (|B|)= \ell$ is shown to be that of the unique strong solution $X$ of the \Ito\ SDE $$ dX_v = \left\{ 4 - X_v^2 \left( t - \mbox{$\int_0^v X_u du$} \right) ^{-1} \right\} \, dv + 2 \sqrt{X_v} d B_v$$ on the interval $[0,V_t (X))$,where $V_t(X):= \inf \{ v: \int_0^v X_u du = t \}$and $X_v = 0$ for all $v \ge V_t(X)$. This conditioned form of the Ray-Knight description of Brownian local times arises from study of the asymptotic distribution as $n \te \infty$ and $ 2 k/\sqrt{n} \te \ell$ of the height profile of a uniform rooted random forest of $k$ trees labeled by a set of $n$ elements, as obtained by conditioning a uniform random mapping of the set to itself to have $k$ cyclic points. The SDE is the continuous analog of a simple description of a Galton-Watson branching process conditioned on its total progeny. A result is obtained regarding the weak convergence of normalizations of such conditioned Galton-Watson processes and height profiles of random forests to a solution of the SDE. For $\ell = 0$, corresponding to asymptotics of a uniform random tree, the SDE gives a new description of the process of local times of a Brownian excursion, implying Jeulin's description of these local times as a time change of twice a Brownian excursion. Another corollary is the Biane-Yor description of the local times of a reflecting Brownian bridge as a time changed reversal of twice a Brownian meander of the same length.