# Best way to count nest depth in e.g. f[f[f[...[f[x]]...]?

What is the best way to count nesting depth in e.g. f[f[f[...[f[x]]...]? In other words, what function func is good for giving you something more or less like the following:

func[Nest[f, h[x], n], f] == n


To be clear, the goal is to specifically count the nesting depth of only one function head (here, f), so f[f[h[f[x]]]] should return 3.

• Depth[#] - 1 & @ Nest[f, x, 7]?
– kglr
Oct 5, 2021 at 20:22
• @Trev kglr was answering the question you asked... Please add that case from your comment to the question, and any other cases you would like handled. Oct 5, 2021 at 20:47

tl;dr: You can get a very fast, and very general (avoiding $RecursionLimit) function by looking for the first position (of the right kind) at which the nesting stops, instead of directly counting how many nestings have occurred. EDIT: However, inspired by MichaelE2's ArrayDepth answer, hdepth[expr_, f_] := Replace[ Dimensions[expr, AllowedHeads -> f], {x : Longest[1 ...], ___} :> Length[{x}]]  seems to be even faster, just as robust, and a lot more readable. But if you'd like to see how looking for an exception to the nesting chain works, read on... While both nice, there are two potential problems with the existing answers: @mikado's nice and clean functional solution will potentially hit $RecursionLimit on very deeply nested f-expressions, e.g. Nest[f, x, 2000], and (as of the time of writing) @kglr's clever identification of Position as a useful tool here only coincides with the "nest depth" in special cases. In general, it gets the depth of the deepest occurrence of f anywhere in the expression; consider f[h[f[x]]].

One option to circumvent the recursion limit is to use ReplaceRepeated (//.) instead of function definitions. There are several ways to make this work, but I tested with

t = Nest[f, x, 50000];


which quickly led to 30-second computation times. I moved Position, but only managed to cut the time in half.

So, here's the trick, using Position, that works quickly:

• Check that the expression is of the form f[_]

• If so, look for the positions of f[patt] where patt matches any sequence of arguments that isn't of the form f[_]

• Take only the positions "on the main branch", i.e. of the form {1, 1, 1, ...}; get their length, and add 1.

For patt, we need to account for the fact that Except can't handle variable-length patterns. So I took patt to be Except[f[_]] | Repeated[_, {0}] | Repeated[_, {2, Infinity}], as I found it to be the fastest among the ones I tried.

hdepth[f_][expr : f_[_]] :=
1 + Min[
Cases[
Position[expr,
f[Except[f[_]] | Repeated[_, {0}] | Repeated[_, {2, Infinity}]]],
x : {1 ...} :> Length[x]]]

hdepth[f_][Except[f_[_]]] := 0


I use two definitions to check that the expression matches f[_], but you could bundle this in a single definition if you want to.

This cuts a 30-second evaluation time down to 0.01 seconds.

hdepth[f][t] // RepeatedTiming

(* {0.01049995313, 50000} *)


Hope this helps; let me know if it misjudges the depth in any case! :)

Here are some tests and pathological cases I could think of; note that since Nest does not have SequenceHold, and thus splices Sequence[...] arguments into its argument sequence, we don't consider f[x,y] or f[] to be a once-nested instance of f.

hdepth[f][x]           (* 0 *)

hdepth[f][f[x]]        (* 1 *)

hdepth[f][f[h[f[x]]]]  (* 1 *)

hdepth[f][f[x][x]]     (* 0 *)

hdepth[f][f[x, y]]     (* 0 *)

hdepth[f][f[]]         (* 0 *)


Note that, admittedly, 50000 is about the most I could Nest f on my computer before crashing the kernel—but I chose this arbitrarily, so I didn't realize until after :)

For fun, here are two that worked, but weren't so fast (despite being the fastest I could find of their "class").

(* Get all "main branch" positions of f[_], then see how many
appear consecutively: *)

hdepth2[expr_, f_] := Catch[Fold[If[#1 === #2, #1 + 1, Throw[#1]] &, 0,
Sort[Cases[Position[expr, f[_]], x : {1 ...} :> Length[x]]]]]

(* Use a module wrapper to prevent accidental matches with //.,
since //. acts on all parts of the expression *)

hdepth3[expr_, f_] :=
Module[{w}, (w[expr,0] //. w[f[x_], n_] :> w[x, n + 1]) // Last]


You can define a simple function to do this:

fdepth[f[u_]] := 1 + fdepth[u]
fdepth[_] = 0;


For example

NestList[f, h[g[x]], 5]
(* {h[g[x]], f[h[g[x]]], f[f[h[g[x]]]], f[f[f[h[g[x]]]]],
f[f[f[f[h[g[x]]]]]], f[f[f[f[f[h[g[x]]]]]]]} *)

fdepth /@ %
(* {0, 1, 2, 3, 4, 5} *)


You can use Position:

ClearAll[nestDepth]
nestDepth[func_] := Max @* Map[Length] @* Position[func]

nestDepth[f] @ Nest[f, w @ g @ h @ x, 8]

8

nestDepth[f] /@ NestList[f, w @ g @ h @ x, 5]

{-∞, 1, 2, 3, 4, 5}


You can also use Count as follows:

ClearAll[nestDepth2]
nestDepth2[func_] := Count[#, Blank @ func, All] &

nestDepth2[f] @ Nest[f, w @ g @ h @ x, 8]

8

nestDepth2[f] /@ NestList[f, w @ g @ h @ x, 5]

{0, 1, 2, 3, 4, 5}


Perhaps ArrayDepth?

ArrayDepth[f[f[f[h[x]]]]]
(*  3  *)

ArrayDepth@Nest[f, x, 7]
(*  7  *)


ArrayDepth has an AllowedHeads option, which is not needed on the OP's example.

• There is one situation where that doesn't capture the number of times f has been nested (on a non-f-headed expression): in situations like f[f[x], f[x]], which has an ArrayDepth of 2 but a "nested depth" of 0. In the same vein, perhaps LengthWhile[Dimensions[t, AllowedHeads -> f], # === 1 &] would work? this is just about as fast as my answer, which is nice! Oct 6, 2021 at 2:10
• @thorimur Thanks. It might depend on the definition of "nested function." Think nested lists: {1, {2}} is often called a nested list. I was concerned rather that f[y, f[x]] might be considered to have a nested depth of 2, instead of 1 as ArrayDepth will return. Hence the remark at the end of my answer. All that aside, I think ArrayDepth is the easiest way to solve the problem the OP presented. Oct 6, 2021 at 2:21
• good point! it's true that the desired behavior is ambiguous; I think I assumed from the presence of Nest in the example that the goal was primarily to "count the number of applications of f" in the kind of expressions produced by Nest, but you're right, that's not necessarily the case. Oct 6, 2021 at 2:33

Interesting answers, I kind of figured there'd be a simpler built-in method for this.

Here's another workable low-tech approach:

fDepth[expr_, f_] := StringCount[ToString@expr, ToString@f <> "["]


Note that without some safeguards, this approach would fail in any context where you have the function f alongside other functions ending with f.