When you multiply two N-bit numbers, you generate an N×N grid of partial-product bits that all need to be summed. The naive approach — ripple down each column — takes O(N) time. Wallace and Dadda trees crush this to O(log N) by using 3:2 compressors (full adders) and 2:2 compressors (half adders) in parallel to reduce the column heights until only two rows remain. Then a single fast adder (Kogge-Stone, usually) finishes the job.

The classic debate: Wallace reduces aggressively — at every stage, it compresses every column that has 3+ bits, using as many full adders as possible. Dadda is lazier — it only reduces columns that must be reduced to hit the next height target in the sequence (2, 3, 4, 6, 9, 13, 19, 28...). Dadda uses fewer half adders and full adders overall, but its final carry-propagate adder is slightly wider. Wallace uses more compressors but a narrower final adder.

In practice, modern designs use hybrid trees. Here's why: full adders and half adders have different delays, area, and power. A half adder is ~half the area of a full adder but only reduces column height by 1 (from 2→1) instead of 2 (from 3→2). Pure Wallace over-uses half adders in the early stages, wasting area. Pure Dadda concentrates work in later stages, creating wire congestion.

Real-world example: The multiplier in Intel's Skylake FMA unit uses a hybrid 4:2 compressor tree (built from two 3:2 compressors with an internal carry). 4:2 compressors have a critical path of only 3 XOR delays regardless of position — flat and regular. For a 53×53 mantissa multiply (double-precision), a pure Wallace tree needs 9 reduction stages; a 4:2 tree needs 6 stages arranged as a regular 2D grid, which places-and-routes beautifully.

Rule of thumb for reduction depth: Starting with column height H, each Wallace stage reduces to ⌈2H/3⌉. To reduce from H down to 2 takes approximately log₁.₅(H/2) stages. For H=27 (a 27×27 multiplier partial product column), that's log₁.₅(13.5) ≈ 6.4, so 7 stages of 3:2 compressors. With 4:2 compressors, halve it: about 4 stages.

The synthesis tool almost always beats hand-crafted Wallace or Dadda because it can mix compressor types per-column based on the actual arrival times of each bit — some partial-product bits arrive later than others due to Booth encoding, so the tree should be skewed to give slower bits shorter paths.