  We formulate a framework for discrete-time quantum walks, motivated by
classical random walks with memory. We present a specific representation of the
classical walk with memory 2 on which this is based. The framework has no need
for coin spaces, it imposes no constraints on the evolution operator other than
unitarity, and is unifying of other approaches. As an example we construct a
symmetric discrete-time quantum walk on the semi-infinite binary tree. The
generating function of the amplitude at the root is computed in closed-form, as
a function of time and the initial level n in the tree, and we find the
asymptotic and a full numerical solution for the amplitude. It exhibits a sharp
interference peak and a power law tail, as opposed to the exponentially
decaying tail of a broadly peaked distribution of the classical symmetric
random walk on a binary tree. The probability peak is orders of magnitude
larger than it is for the classical walk (already at small n). The quantum walk
shows a polynomial algorithmic speedup in n over the classical walk, which we
conjecture to be of the order 2/3, based on strong trends in data.
