diff --git a/src/functor.jl b/src/functor.jl
index 267ad42..2956524 100644
--- a/src/functor.jl
+++ b/src/functor.jl
@@ -26,7 +26,7 @@ function makefunctor(m::Module, T, fs = fieldnames(T))
   escargs = map(fieldnames(T)) do f
     f in fs ? :(y[$(yᵢ += 1)]) : :(x.$f)
   end
-  escfs = [:($f=x.$f) for f in fs]
+  escfs = [:($f = x.$f) for f in fs]
 
   @eval m begin
     $Functors.functor(::Type{<:$T}, x) = ($(escfs...),), y -> $T($(escargs...))
@@ -169,7 +169,54 @@ function _default_walk(f, x)
   func, re = functor(x)
   re(map(f, func))
 end
-_default_walk(f, ::Nothing, ::Nothing) = nothing
+_default_walk(_, ::Nothing, ::Nothing) = nothing
+
+# Side effects only, saves a restructure
+function _foreach_walk(f, x)
+  foreach(f, children(x))
+  return x
+end
+
+### WARNING: the following is unstable internal functionality. Use at your own risk!
+# Wrapper over an IdDict which only saves values with a stable object identity
+struct Cache{K,V}
+  inner::IdDict{K,V}
+end
+Cache() = Cache(IdDict())
+
+usecache(x) = !isbits(x) && ismutable(x)
+# Functionally immutable and observe value semantics, but still `ismutable` and not `isbits`
+usecache(::Union{String,Symbol}) = false
+# For varargs
+usecache(xs::Tuple) = all(usecache, xs)
+Base.get!(f, c::Cache, x) = usecache(x) ? get!(f, c.inner, x) : f()
+
+# Passthrough used to disable caching (e.g. when passing `cache=false`)
+struct NoCache end
+Base.get!(f, ::NoCache, _) = f()
+
+# Encapsulates the self-recursive part of a recursive tree reduction (fold).
+# This allows calling functions to remove any self-calls or nested callback closures.
+struct Fold{F,L,C,W}
+  fn::F
+  isleaf::L
+  cache::C
+  walk::W
+end
+(fld::Fold)(x) = get!(fld.cache, x) do 
+  fld.fn(fld.isleaf(x) ? x : fld.walk(fld, x))
+end
+
+# Convenience function for working with `Fold`
+function fold(f, x; isleaf = isleaf, cache = false, walk = _default_walk)
+  if cache === true
+    cache = Cache()
+  elseif cache === false
+    cache = NoCache()
+  end
+  return Fold(f, isleaf, cache, walk)(x)
+end
+### end of unstable internal functionality
 
 """
     fmap(f, x; exclude = Functors.isleaf, walk = Functors._default_walk)
@@ -253,11 +300,9 @@ Foo(Bar([1, 2, 3]), (40, 50, Bar(Foo(6, 7))))
 ```
 """
 function fmap(f, x; exclude = isleaf, walk = _default_walk, cache = IdDict())
-  haskey(cache, x) && return cache[x]
-  y = exclude(x) ? f(x) : walk(x -> fmap(f, x, exclude = exclude, walk = walk, cache = cache), x)
-  cache[x] = y
-
-  return y
+  return fold(x; cache, walk, isleaf = exclude) do node
+    exclude(node) ? f(node) : node
+  end
 end
 
 """
@@ -296,8 +341,8 @@ fmapstructure(f, x; kwargs...) = fmap(f, x; walk = (f, x) -> map(f, children(x))
     fcollect(x; exclude = v -> false)
 
 Traverse `x` by recursing each child of `x` as defined by [`functor`](@ref)
-and collecting the results into a flat array, ordered by a breadth-first
-traversal of `x`, respecting the iteration order of `children` calls.
+and collecting the results into a flat array, ordered by a depth-first,
+post-order traversal of `x` that respects the iteration order of `children` calls.
 
 Doesn't recurse inside branches rooted at nodes `v`
 for which `exclude(v) == true`.
@@ -324,33 +369,27 @@ Foo(Bar([1, 2, 3]), NoChildren(:a, :b))
 
 julia> fcollect(m)
 4-element Vector{Any}:
- Foo(Bar([1, 2, 3]), NoChildren(:a, :b))
- Bar([1, 2, 3])
  [1, 2, 3]
+ Bar([1, 2, 3])
  NoChildren(:a, :b)
+ Foo(Bar([1, 2, 3]), NoChildren(:a, :b))
 
 julia> fcollect(m, exclude = v -> v isa Bar)
 2-element Vector{Any}:
- Foo(Bar([1, 2, 3]), NoChildren(:a, :b))
  NoChildren(:a, :b)
+ Foo(Bar([1, 2, 3]), NoChildren(:a, :b))
 
 julia> fcollect(m, exclude = v -> Functors.isleaf(v))
 2-element Vector{Any}:
- Foo(Bar([1, 2, 3]), NoChildren(:a, :b))
  Bar([1, 2, 3])
+ Foo(Bar([1, 2, 3]), NoChildren(:a, :b))
 ```
 """
-function fcollect(x; output = [], cache = Base.IdSet(), exclude = v -> false)
-    # note: we don't have an `OrderedIdSet`, so we use an `IdSet` for the cache
-    # (to ensure we get exactly 1 copy of each distinct array), and a usual `Vector`
-    # for the results, to preserve traversal order (important downstream!).
-    x in cache && return output
-    if !exclude(x)
-      push!(cache, x)
-      push!(output, x)
-      foreach(y -> fcollect(y; cache=cache, output=output, exclude=exclude), children(x))
-    end
-    return output
+function fcollect(x; output = [], cache = Base.IdDict(), exclude = v -> false)
+  fold(x; cache, isleaf = exclude, walk = _foreach_walk) do node
+    exclude(node) || push!(output, node);  # always return nothing
+  end
+  return output
 end
 
 # Allow gradients and other constructs that match the structure of the functor
diff --git a/test/basics.jl b/test/basics.jl
index 21aa445..ad4e06a 100644
--- a/test/basics.jl
+++ b/test/basics.jl
@@ -31,6 +31,26 @@ end
   end
 end
 
+@testset "Folds" begin
+  arrays = ntuple(i -> [i], 3)
+  model = Foo(
+    Foo(arrays[1], arrays[2]),
+    Foo(arrays[3], arrays[1])
+  )
+
+  total = Ref(0)
+  Functors.fmap(model, cache = true) do x
+    total[] += only(x)
+  end
+  @test total[] == 6
+
+  total = Ref(0)
+  Functors.fmap(model, cache = false) do x
+    total[] += only(x)
+  end
+  @test total[] == 7
+end
+
 @testset "Nested" begin
   model = Bar(Foo(1, [1, 2, 3]))
 
@@ -72,8 +92,8 @@ end
   m2 = 1
   m3 = Foo(m1, m2)
   m4 = Bar(m3)
-  @test all(fcollect(m4) .=== [m4, m3, m1, m2])
-  @test all(fcollect(m4, exclude = x -> x isa Array) .=== [m4, m3, m2])
+  @test all(fcollect(m4) .=== [m1, m2, m3, m4])
+  @test all(fcollect(m4, exclude = x -> x isa Array) .=== [m2, m3, m4])
   @test all(fcollect(m4, exclude = x -> x isa Foo) .=== [m4])
 
   m1 = [1, 2, 3]
@@ -81,12 +101,12 @@ end
   m0 = NoChildren(:a, :b)
   m3 = Foo(m2, m0)
   m4 = Bar(m3)
-  @test all(fcollect(m4) .=== [m4, m3, m2, m1, m0])
+  @test all(fcollect(m4) .=== [m1, m2, m0, m3, m4])
 
   m1 = [1, 2, 3]
   m2 = [1, 2, 3]
   m3 = Foo(m1, m2)
-  @test all(fcollect(m3) .=== [m3, m1, m2])
+  @test all(fcollect(m3) .=== [m1, m2, m3])
 end
 
 struct FFoo
@@ -143,8 +163,8 @@ end
   m2 = [1, 2, 3]
   m3 = FFoo(m1, m2, (:y, ))
   m4 = FBar(m3, (:x,))
-  @test all(fcollect(m4) .=== [m4, m3, m2])
-  @test all(fcollect(m4, exclude = x -> x isa Array) .=== [m4, m3])
+  @test all(fcollect(m4) .=== [m2, m3, m4])
+  @test all(fcollect(m4, exclude = x -> x isa Array) .=== [m3, m4])
   @test all(fcollect(m4, exclude = x -> x isa FFoo) .=== [m4])
 
   m0 = NoChildren(:a, :b)
@@ -152,5 +172,5 @@ end
   m2 = FBar(m1, ())
   m3 = FFoo(m2, m0, (:x, :y,))
   m4 = FBar(m3, (:x,))
-  @test all(fcollect(m4) .=== [m4, m3, m2, m0])
+  @test all(fcollect(m4) .=== [m2, m0, m3, m4])
 end