  We study the problem of learning properties of nodes in tree structures.
Those properties are specified by logical formulas, such as formulas from
first-order or monadic second-order logic. We think of the tree as a database
encoding a large dataset and therefore aim for learning algorithms which depend
at most sublinearly on the size of the tree. We present a learning algorithm
for quantifier-free formulas where the running time only depends polynomially
on the number of training examples, but not on the size of the background
structure. By a previous result on strings we know that for general first-order
or monadic second-order (MSO) formulas a sublinear running time cannot be
achieved. However, we show that by building an index on the tree in a linear
time preprocessing phase, we can achieve a learning algorithm for MSO formulas
with a logarithmic learning phase.
