# usainboltz.examples.polya_trees¶

A Boltzmann sampler for Pólya n-ary trees.

>>> from usainboltz import *
>>> from usainboltz.generator import rng_seed


Pólya trees are unlabelled unordered rooted trees. They are specified using the MSet operator:

>>> z = Atom()
>>> T = RuleName("T")
>>> grammar = Grammar({T: z * MSet(T)})
>>> grammar
{
T : Product(z, MSet(T))
}


The singularity of theses trees is near z = 0.33832185689920769 (cf. [FS2009] section VII.5 page 475) and the value of the generating function at this point is 1. Let us check that the oracle finds these values:

>>> oracle = build_oracle(grammar)
>>> values = oracle.tuning(z, singular=True)
>>> # NB: values[z][j] stores the value of z^j
>>> print(f"{values[z]:.6f}")
0.338322
>>> print(f"{values[T]:.6f}")
1.000000


Now, let us generate a Pólya tree: >>> generator = Generator(grammar) >>> res = generator.sample((10, 15)) >>> print(res.obj) (‘z’, [(‘z’, []),

(‘z’, [(‘z’, []),

(‘z’, []), (‘z’, [(‘z’, []),

(‘z’, [(‘z’, []),
(‘z’, []), (‘z’, []), (‘z’, []), (‘z’, [])])])])])

Finally, let us check that the number of atoms in the structure corresponds to the size reported by the generator. >>> def compute_size(tree): … _, mset = tree … return 1 + sum((compute_size(tree) for tree in mset)) >>> res.sizes[z] == compute_size(res.obj), res.sizes[z] (True, 13)