E : $e_k$ is the number of children of node $k$ of the forest associated with the binary tree.

LR : two $n$-arrays, $l$ and $r$, specify the left and right children; $l_k$ ($r_k$) is the left (right) child of node $k$. This is the format output by BinaryTrees.

S : $s_k$ is the number of descendants of node $k$ of the forest associated with the binary tree.

A : binary representation of n pairs of nested parentheses. The array is indexed from 1 to 2n. A zero (one) at index k corresponds to a left (right) parenthesis. This is the format output by NestedParentheses.

C : inversion table encoding of the permutation representation (P). This also corresponds to a level encoding of the corresponding forest; $c_k$ is the level of the forest's $k$th node in preorder.

D : run-length encoding.  A string of nested-parentheses is encoded as ${\left(\right)}^{d_1}{\left(\right)}^{d_2}\mathrm{...}{\left(\right)}^{d_n}$, where the exponent $d_k$ produces $d_k$ nested parentheses.

P : permutation encoding. The permutation $p_1,p_2,\mathrm{...},p_n$ represents a parenthesis string, where the $k$th right parenthesis matches the $p_k$th left parenthesis.

Z : $z_k$ is the location of the $k$th left parenthesis.

Construct an iterator that generates all binary trees with four internal nodes, in LR format.

$B≔\mathrm{BinaryTrees}\left(4\right)\:$

Print the LR, C, E, and S representations of the binary trees.

$\mathrm{printf}\left(''L\; |\; R\; |\; C\; |\; E\; |\; S''\right)\:$$\mathbf{for}\mathrm{lr}\mathbf{in}B\mathbf{do}c≔\mathrm{Convert}\left(\mathrm{lr}\,\'\mathrm{LR}\'\,\'C\'\right)\;s≔\mathrm{Convert}\left(c\,\'C\'\,\'S\'\right)\;e≔\mathrm{Convert}\left(s\,\'S\'\,\'E\'\right)\;\mathrm{printf}\left(''\%\{\}d\; |\; \%\{\}d\; |\; \%\{\}d\; |\; \%\{\}d\; |\; \%\{\}d''\,\mathrm{lr}\,c\,e\,s\right)\mathbf{end\; do}\:$

   L    |    R    |    C    |    E    |    S

2 3 4 0 | 0 0 0 0 | 0 1 2 3 | 1 1 1 0 | 3 2 1 0
0 3 4 0 | 2 0 0 0 | 0 0 1 2 | 0 1 1 0 | 0 2 1 0
2 0 4 0 | 0 3 0 0 | 0 1 1 2 | 2 0 1 0 | 3 0 1 0
2 0 4 0 | 3 0 0 0 | 0 1 0 1 | 1 0 1 0 | 1 0 1 0
0 0 4 0 | 2 3 0 0 | 0 0 0 1 | 0 0 1 0 | 0 0 1 0
2 3 0 0 | 0 0 4 0 | 0 1 2 2 | 1 2 0 0 | 3 2 0 0
0 3 0 0 | 2 0 4 0 | 0 0 1 1 | 0 2 0 0 | 0 2 0 0
2 3 0 0 | 0 4 0 0 | 0 1 2 1 | 2 1 0 0 | 3 1 0 0
2 3 0 0 | 4 0 0 0 | 0 1 2 0 | 1 1 0 0 | 2 1 0 0
0 3 0 0 | 2 4 0 0 | 0 0 1 0 | 0 1 0 0 | 0 1 0 0
2 0 0 0 | 0 3 4 0 | 0 1 1 1 | 3 0 0 0 | 3 0 0 0
2 0 0 0 | 4 3 0 0 | 0 1 1 0 | 2 0 0 0 | 2 0 0 0
2 0 0 0 | 3 0 4 0 | 0 1 0 0 | 1 0 0 0 | 1 0 0 0
0 0 0 0 | 2 3 4 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0

Construct an iterator that generates all nested pairs of four parentheses.

$A≔\mathrm{NestedParentheses}\left(4\right)\:$

Print the string of parentheses and its A, C, D, P, and Z representations.

$\mathrm{printf}\left(''|\; A\; |\; C\; |\; D\; |\; P\; |\; Z''\right)\:$$\mathbf{for}a\mathbf{in}A\mathbf{do}z≔\mathrm{Convert}\left(a\,\'A\'\,\'Z\'\right)\;c≔\mathrm{Convert}\left(z\,\'Z\'\,\'C\'\right)\;d≔\mathrm{Convert}\left(z\,\'Z\'\,\'\mathrm{D}\'\right)\;p≔\mathrm{Convert}\left(c\,\'C\'\,\'P\'\right)\;\mathrm{printf}\left(''\%s\; |\; \%\{\}d\; |\; \%\{\}d\; |\; \%\{\}d\; |\; \%\{\}d\; |\; \%\{\}d''\,\mathrm{String}\left(A\right)\,a\,c\,d\,p\,z\right)\mathbf{end\; do}\:$

         |        A        |    C    |    D    |    P    |    Z

NestedParentheses(4) | 0 1 0 1 0 1 0 1 | 0 0 0 0 | 1 1 1 1 | 1 2 3 4 | 1 3 5 7
NestedParentheses(4) | 0 1 0 1 0 0 1 1 | 0 0 0 1 | 1 1 0 2 | 1 2 4 3 | 1 3 5 6
NestedParentheses(4) | 0 1 0 0 1 1 0 1 | 0 0 1 0 | 1 0 2 1 | 1 3 2 4 | 1 3 4 7
NestedParentheses(4) | 0 1 0 0 1 0 1 1 | 0 0 1 1 | 1 0 1 2 | 1 3 4 2 | 1 3 4 6
NestedParentheses(4) | 0 1 0 0 0 1 1 1 | 0 0 1 2 | 1 0 0 3 | 1 4 3 2 | 1 3 4 5
NestedParentheses(4) | 0 0 1 1 0 1 0 1 | 0 1 0 0 | 0 2 1 1 | 2 1 3 4 | 1 2 5 7
NestedParentheses(4) | 0 0 1 1 0 0 1 1 | 0 1 0 1 | 0 2 0 2 | 2 1 4 3 | 1 2 5 6
NestedParentheses(4) | 0 0 1 0 1 1 0 1 | 0 1 1 0 | 0 1 2 1 | 2 3 1 4 | 1 2 4 7
NestedParentheses(4) | 0 0 1 0 1 0 1 1 | 0 1 1 1 | 0 1 1 2 | 2 3 4 1 | 1 2 4 6
NestedParentheses(4) | 0 0 1 0 0 1 1 1 | 0 1 1 2 | 0 1 0 3 | 2 4 3 1 | 1 2 4 5
NestedParentheses(4) | 0 0 0 1 1 1 0 1 | 0 1 2 0 | 0 0 3 1 | 3 2 1 4 | 1 2 3 7
NestedParentheses(4) | 0 0 0 1 1 0 1 1 | 0 1 2 1 | 0 0 2 2 | 3 2 4 1 | 1 2 3 6
NestedParentheses(4) | 0 0 0 1 0 1 1 1 | 0 1 2 2 | 0 0 1 3 | 3 4 2 1 | 1 2 3 5
NestedParentheses(4) | 0 0 0 0 1 1 1 1 | 0 1 2 3 | 0 0 0 4 | 4 3 2 1 | 1 2 3 4