Rémy trees · MolFoundry

1 · Rules & state space

Insertion with the old subtree on the left — At a selected edge, insert a binary vertex and a leaf. The old subtree goes left or right; each choice has rate κ. This is a tree schematic; the DPO encoding uses additional vertices and edge types. View DPO rule in MolFoundry.

Minimal tree with a left spine of three internal vertices — A minimal tree with P3=1. P3 sums the I, L and R typed-stem pattern counts.

Tree growth and multiplicities

All planar trees through three insertions. A label ×m means m distinct rule matches reach the same tree; the transition rate is mκ. Select a tree to open its incidence graph in MolFoundry.

Inspect individual trees

Insertions n

2 · Inputs & semantics

Six rules and their graph language

Finite rooted planar full binary trees, with an exterior stem. I/L/R edge types encode the root, left and right child incidences. The six DPO rules implement the insertions of the SqPO source model on this language; deleted edge-nodes have exactly their matched incidences.

Insertion schematic GL_I — GL_I · View rule in MolFoundry

Insertion schematic GL_L — GL_L · View rule in MolFoundry

Insertion schematic GL_R — GL_R · View rule in MolFoundry

Insertion schematic GR_I — GR_I · View rule in MolFoundry

Insertion schematic GR_L — GR_L · View rule in MolFoundry

Insertion schematic GR_R — GR_R · View rule in MolFoundry

The pictures show tree insertions; the links open the DPO rules in MolFoundry.

3 · Observable discovery

Discovered patterns and composition steps

The drawings are minimal tree witnesses; each coordinate sums three typed-stem identities. The Pk labels describe the discovered graph shapes.

Minimal tree witness for P3 — P3 · full ordered-binary left-spine pattern of depth 3
P3_I · P3_L · P3_R

Minimal tree witness for P2 — P2 · full ordered-binary left-spine pattern of depth 2
P2_I · P2_L · P2_R

Minimal tree witness for P1 — P1 · full ordered-binary left-spine pattern of depth 1
P1_I · P1_L · P1_R

Minimal tree witness for E — E · all selectable directed typed edges
E_I · E_L · E_R

4 · ODE & pJC

Seed P3 → discovered P2, P1, E. The four-dimensional expectation system closes. Full event partition ↓

Derivative	Computed right-hand side	New coordinate
d E[P3]/dt	κ E[-4 P3 + 4 P2]	P2
d E[P2]/dt	κ E[-2 P2 + 3 P1]	P1
d E[P1]/dt	κ E[2 E]	E
d E[E]/dt	κ E[4 E]	—

Full event partition

The full event partition additionally uses the counting identity N(r,s)=P_(r+s). The expectation ODE follows from composition and tree-count reduction.

Class	Δ(E,P1,P2,P3)	Rate / κ
J0	[2, 1, 0, 0]	2E - 3P1 + 2*P2 - P3
J1	[2, 1, 1, 0]	3P1 - 6P2 + 3*P3
J2	[2, 1, 1, 1]	4P2 - 3P3
J3	[2, 1, 0, -1]	P3

5 · Initial states

Start from one leaf and its exterior stem: E=1 and P1=P2=P3=0. All six rules share κ, called rho in the input.

6 · Dynamics

Common rate κ

Numerical integration of the generated linear coefficients; no finite-state truncation.

Condition on the number of insertions

Here n is the number of insertion events, not elapsed time. These distributions propagate the generated event weights with exact integers.

Insertions n