Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000523
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Mapping
St000523: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => 0
[[],[]]
 => 0
[[[]]]
 => 1
[[],[],[]]
 => 0
[[],[[]]]
 => 0
[[[]],[]]
 => 0
[[[],[]]]
 => 1
[[[[]]]]
 => 1
[[],[],[],[]]
 => 0
[[],[],[[]]]
 => 0
[[],[[]],[]]
 => 0
[[],[[],[]]]
 => 0
[[],[[[]]]]
 => 1
[[[]],[],[]]
 => 0
[[[]],[[]]]
 => 1
[[[],[]],[]]
 => 0
[[[[]]],[]]
 => 1
[[[],[],[]]]
 => 1
[[[],[[]]]]
 => 1
[[[[]],[]]]
 => 1
[[[[],[]]]]
 => 1
[[[[[]]]]]
 => 1
[[],[],[],[],[]]
 => 0
[[],[],[],[[]]]
 => 0
[[],[],[[]],[]]
 => 0
[[],[],[[],[]]]
 => 0
[[],[],[[[]]]]
 => 1
[[],[[]],[],[]]
 => 0
[[],[[]],[[]]]
 => 0
[[],[[],[]],[]]
 => 0
[[],[[[]]],[]]
 => 1
[[],[[],[],[]]]
 => 0
[[],[[],[[]]]]
 => 0
[[],[[[]],[]]]
 => 0
[[],[[[],[]]]]
 => 1
[[],[[[[]]]]]
 => 1
[[[]],[],[],[]]
 => 0
[[[]],[],[[]]]
 => 0
[[[]],[[]],[]]
 => 0
[[[]],[[],[]]]
 => 1
[[[]],[[[]]]]
 => 2
[[[],[]],[],[]]
 => 0
[[[[]]],[],[]]
 => 1
[[[],[]],[[]]]
 => 1
[[[[]]],[[]]]
 => 2
[[[],[],[]],[]]
 => 0
[[[],[[]]],[]]
 => 0
[[[[]],[]],[]]
 => 0
[[[[],[]]],[]]
 => 1
[[[[[]]]],[]]
 => 1
Description
The number of 2-protected nodes of a rooted tree. This is the number of nodes with minimal distance two to a leaf. The number of trees with no 2-protected nodes is [[oeis:A143363]].