OneProtectedNodeCount

The number of vertices whose nearest leaf is exactly one edge away.

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OneProtectedNodeCount(tree)The number of vertices whose nearest leaf is exactly one edge away.

Typed signatures

Type	Package	Applies
(ordered_tree) -> number	combinatorics	always

Domain: Tree statistics

Details

• Defined over OrderedTree as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a OrderedTree and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on OrderedTree, over _x

Apply(
       Function([0, 0] if Length(t) == 0 else Apply(
                                                     (r: collection) => [r[1] +
                                                                         (1 if 1 +
                                                                               r[2] ==
                                                                               1 else 0), 1 +
                                                                           r[2]],
                                                     Fold(
                                                           Function([
                                                                      a[1] +
                                                                      b[1],
                                                                      Min(
                                                                           a[2],
                                                                           b[2],
                                                                         ),
                                                                    ], a, b),
                                                           [0, PositiveInfinity],
                                                           Map(Function(
                                                                         self(self, c),
                                                                         c,
                                                                       ), t),
                                                         ),
                                                   ), self, t),
       Function([0, 0] if Length(t) == 0 else Apply(
                                                     (r: collection) => [r[1] +
                                                                         (1 if 1 +
                                                                               r[2] ==
                                                                               1 else 0), 1 +
                                                                           r[2]],
                                                     Fold(
                                                           Function([
                                                                      a[1] +
                                                                      b[1],
                                                                      Min(
                                                                           a[2],
                                                                           b[2],
                                                                         ),
                                                                    ], a, b),
                                                           [0, PositiveInfinity],
                                                           Map(Function(
                                                                         self(self, c),
                                                                         c,
                                                                       ), t),
                                                         ),
                                                   ), self, t),
       _x,
     )[1]

Examples

Basic #

the ordered tree $[[],[[]],[]]$

live ↗

In$\mathrm{OneProtectedNodeCount}({\operatorname{OrderedTree}}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack))$Out$2$

Properties #

FindStat's value for [[]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{OrderedTree}}(\bigl\lbrack\bigl\lbrack\bigr\rbrack\bigr\rbrack), \text{OneProtectedNodeCount})$Out$1$

Known value · FindStat St000522

FindStat's value for [[[]]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{OrderedTree}}(\bigl\lbrack\bigl\lbrack\bigl\lbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack), \text{OneProtectedNodeCount})$Out$1$

Known value · FindStat St000522

FindStat's value for [[],[]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{OrderedTree}}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack), \text{OneProtectedNodeCount})$Out$1$

Known value · FindStat St000522

About

Examples	4
Crosswalk	FindStat