# Mapping on leaves

Say I have some nested structure, such as {a,{{b,c},d,{e,{f,g}}}}, and I want to apply a function $q$ to each of the leaves; that is, I want the output to be {q[a],{{q[b],q[c]},q[d],{q[e],{q[f],q[g]}}}}. There must be a primitive to do this, but I can't find it. I initially thought that Map with a third argument of Infinity would do it, but that does something different (in addition to applying q at the leaves, it also applies it to each higher-level list element).

• Attributes[q] = Listable or you can Map at {-1} unless a,b,c... are non atomic,
– Kuba
Commented Nov 16, 2015 at 14:53
• @Kuba I guess I should have read that page more carefully. Thanks. Commented Nov 16, 2015 at 14:56
• @Kuba I should note that the Attributes[q] = Listable version won't work unless all containers for leaves are Lists. Commented Nov 16, 2015 at 15:53

A little "secret" of level specifications is that they can be negative. -1 refers to the atomic leaves, -2 refers to all Depth 2 subexpressions, generally -k refers to all depth k subexpressions. Thus the behaviour of negative levels is somewhat different from that of positive ones.

Mapping at level {-1} (i.e. only level -1, not a range of levels) will accomplish what you need.

Level[{a, {{b, c}, d, {e, {f, g}}}}, {-1}]
(* {a, b, c, d, e, f, g} *)

Map[x, {a, {{b, c}, d, {e, {f, g}}}}, {-1}]
(* {x[a], {{x[b], x[c]}, x[d], {x[e], {x[f], x[g]}}}} *)

• This reminds me of Bob Ross [badly paraphrased]: "...just happy little secrets..." Commented Nov 16, 2015 at 15:09
• Worth linking: Levels: how do they work?
– Kuba
Commented Nov 16, 2015 at 15:31
• @Kuba Thanks! I knew there had to be such a post, but couldn't find the right one. I wasn't familiar with this specific question you linked, but it seems to be the best one. Commented Nov 16, 2015 at 15:47