  Branched covers of the complex projective line ramified over $0,1$ and
$\infty$ (Grothendieck's {\em dessins d'enfant}) of fixed genus and degree are
effectively enumerated. More precisely, branched covers of a given ramification
profile over $\infty$ and given numbers of preimages of $0$ and $1$ are
considered. The generating function for the numbers of such covers is shown to
satisfy a PDE that determines it uniquely modulo a simple initial condition.
Moreover, this generating function satisfies an infinite system of PDE's called
the KP (Kadomtsev-Petviashvili) hierarchy. A specification of this generating
function for certain values of parameters generates the numbers of {\em
dessins} of given genus and degree, thus providing a fast algorithm for
computing these numbers.
