Getting Started

This page walks through the basics of kauri: creating trees, inspecting their properties, computing coproducts, and checking Runge–Kutta order conditions.

Creating and displaying a tree

A rooted tree is built from nested lists. The empty list [] represents a leaf (single vertex), and nesting adds branches.

import kauri as kr

t = kr.Tree([[], [[]]])
kr.display(t)

Tree properties

Every tree carries combinatorial data used in B-series and order condition theory.

import kauri as kr

t = kr.Tree([[], [[]]])
print("nodes  :", t.nodes())
print("factorial:", t.factorial())
print("sigma  :", t.sigma())
print("density :", t.density())
print("alpha  :", t.alpha())

nodes  : 4
factorial: 8
sigma  : 1
density : 0.3333333333333333
alpha  : 3

BCK coproduct

The Butcher–Connes–Kreimer coproduct decomposes a tree into admissible cuts.

import kauri as kr
import kauri.bck as bck

t = kr.Tree([[], [[]]])
kr.display(bck.coproduct(t))

Runge–Kutta methods

Kauri ships with several predefined Runge–Kutta methods. You can query their order and inspect elementary weights.

import kauri as kr
from kauri.rk_methods import rk4

print("RK4 order:", rk4.order())
ew = rk4.elementary_weights_map()
for t in kr.trees_of_order(4):
    print(repr(t), "weight:", ew(t))

RK4 order: 4
[[[[]]]] weight: 0.041666666666666664
[[[], []]] weight: 0.08333333333333333
[[[]], []] weight: 0.125
[[], [], []] weight: 0.25

Enumerating trees

Kauri can generate all rooted trees of a given order.

import kauri as kr

for t in kr.trees_of_order(4):
    kr.display(t)

Forest arithmetic

Multiplying trees produces forests, and adding them produces formal linear combinations (forest sums).

import kauri as kr

t1 = kr.Tree([])
t2 = kr.Tree([[]])
forest = t1 * t2
print("Forest:", repr(forest))

forest_sum = t1 + t2 - 2 * kr.Tree([[], []])
print("ForestSum:", repr(forest_sum))

Forest: [] [[]]
ForestSum: 1 * [] + 1 * [[]] + -2 * [[], []]

Planar trees

Planar trees preserve sibling order. Two trees that are equal in the non-planar setting can be distinct when planar.

import kauri as kr

t1 = kr.PlanarTree([[], [[]]])
t2 = kr.PlanarTree([[[]], []])
print("t1 == t2:", t1 == t2)
kr.display(t1)
kr.display(t2)

t1 == t2: False