1 | initial version |

The trick is probably to define correctly what is a leaf. First rough attempt :

```
def LC(x):
def is_leaf(x):
from sage.symbolic.expression import Expression
return type(x) is not Expression or x.is_symbol() or x.is_numeric()
if is_leaf(x): return 1
return sum(map(LC, x.operands()))
sage: LC(sin(a+b))
2
sage: LC(sin(a+b).trig_expand())
4
sage: LC(integrate(sin(x), x, a, b, hold=True))
4
```

Note that, by design, it won't work on anything else than a symbolic expression ; polynomials, rational fractions, series, etc..., here considered as a leaf, need their own implementation, whose definition of "leaf" isn't a trivial question. Consider :

```
sage: R1.<t>=QQ[]
sage: LC(t^3-3*t^2+t-1)
1
sage: LC(SR(t^3-3*t^2+t-1))
7
sage: len((t^3-3*t^2+t-1).coefficients())
4
sage: len((t^3).coefficients())
1
```

Anyway, this is probably not sufficiently "general" to deserve an implementation in "core" Sage. A package grouping some tree-related utilities could be considered...

BTW, the "complexity" of an expression isn't only a function of the number of its leaves, but also of the number of itsnon-terminal nodes, the distribution of the depth of its branches and the structure of common subexpressions... AFAIK, there is no universally acknowledged metric of an expression's "complexity"...

HTH,

2 | No.2 Revision |

The trick is probably to define correctly what is a leaf. First rough attempt :

```
def LC(x):
def is_leaf(x):
from sage.symbolic.expression import Expression
return type(x) is not Expression or x.is_symbol() or x.is_numeric()
if is_leaf(x): return 1
return sum(map(LC, x.operands()))
sage: LC(sin(a+b))
2
sage: LC(sin(a+b).trig_expand())
4
sage: LC(integrate(sin(x), x, a, b, hold=True))
4
```

Note that, by design, it won't work on anything else than a symbolic expression ; polynomials, rational fractions, series, etc..., here considered as a leaf, need their own implementation, whose definition of "leaf" isn't a trivial question. Consider :

```
sage: R1.<t>=QQ[]
sage: LC(t^3-3*t^2+t-1)
1
sage: LC(SR(t^3-3*t^2+t-1))
7
sage: len((t^3-3*t^2+t-1).coefficients())
4
sage: len((t^3).coefficients())
1
sage: len((t^3).coefficients(sparse=False))
4
```

Anyway, this is probably not sufficiently "general" to deserve an implementation in "core" Sage. A package grouping some tree-related utilities could be considered...

BTW, the "complexity" of an expression isn't only a function of the number of its leaves, but also of the number of ~~itsnon-terminal ~~its non-terminal nodes, the distribution of the depth of its branches and the structure of common subexpressions... AFAIK, there is no universally acknowledged metric of an expression's "complexity"...

HTH,

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