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A tree representation of a list with empty nodes, single-element nodes and binary concatenation nodes. Concatenation constructs one node without traversing either operand. The representation need not be unique: adding empty branches changes its size without changing the denoted list. An accumulator-based traversal gives linear flattening of a binary tree in the representation's size rather than merely the output length.

Carry the eventual output tail as an accumulator, thereby avoiding repeated list concatenation:

abs :: CatList a -> [a]
abs t = flatten t []
  where
    flatten Nil       zs = zs
    flatten (One x)   zs = x : zs
    flatten (Cat l r) zs = flatten l (flatten r zs)

The invariant is flatten t zs = denotation t ++ zs. It is immediate for Nil and One; at Cat, applying the two induction hypotheses and associativity of list concatenation proves it. In particular, the empty accumulator gives the required abstraction.

Each concatenation tree constructor is processed once, and each One emits one result cons. Consequently full production takes $\Theta (m)$ time, where $m$ is the input constructor count, on every tree shape. This is a difference list calculation written with an explicit tail, an instance of linear flattening of a binary tree. The call stack can have depth $m$ for a badly skewed input; a linear time bound does not imply logarithmic stack space.

Let $n$ be the number of leaves; the total number of nodes is $2n-1$. List concatenation copies its left operand. For a tree whose left subtree contains $n-1$ leaves at each successive internal node, these copied lengths are $1,2,\dots ,n-1$. Their sum is $n(n-1)/2$, so the worst-case running time is $\Theta (n^2)$.

A difference list representation avoids copying prefixes:

flattenFast :: Tree a -> [a]
flattenFast t = go t []
  where
    go (One x)   acc = x:acc
    go (Two l r) acc = go l (go r acc)

The recursion invariant is go t acc = flatten t ++ acc. The leaf case is immediate; the internal-node case uses associativity of list concatenation. Each node is visited once and each leaf contributes one output constructor, giving $\Theta (n)$ running time when the result is fully consumed. This is linear flattening of a binary tree and does not assume balance.