VectorPaint prefix-code trees: put four symbols at leaves and decode bit by bit

English translation of the Chinese original, translated on 2026-11-13. Original publication dates, attribution, code and version notes are retained.

Draw a binary coding tree for four symbols, matching edge labels, leaf symbols and root-to-leaf paths. This explains prefix coding without making VectorPaint a compressor or claiming an invented code is optimal for arbitrary data. Check coding, decoding and layout separately.

Declare four symbols and their codewords

On a 1200×1600 portrait canvas first declare A=0, B=10, C=110 and D=111 on paper. The NIST prefix-code definition requires that no complete codeword prefix another. Bits label edges; A–D are symbols, not a binary value attached to the root. Separate nodes, edges, bits, symbols and audit paths. Fix the full codebook before reading any message rather than silently changing it during decoding.

Split the root into a zero leaf and one branch

Place root R above a left edge labelled 0 ending at leaf A, and a right edge labelled 1 ending at internal node I. Leave label space and attach bits to their own edge midpoints. A has no descendants because 0 already completes its codeword; placing B below A would alter prefix relations. Use available path controls; arrows are optional if top-to-bottom traversal is declared. Decorative root circles must not create extra levels or bits.

Continue the one branch to B and the final pair

From I draw 0 left to B and 1 right to internal J. From J draw 0 left to C and 1 right to D. Root paths are 10, 110 and 111, not just the final nearby edge. Count four leaves A–D, three internal nodes R/I/J and six edges: seven nodes are not seven codewords. The NIST coding-tree entry explains root-to-leaf code paths; a tree-like drawing does not authorise symbol output at every internal node.

Check prefixes using complete codewords

Only A begins with 0, while the others begin with 1, so A prefixes none of them. B’s 10 differs from C/D’s 11 at the second bit, and 110 versus 111 differ at the third. Check pairs explicitly; unequal lengths alone prove neither conflict nor validity. Coloured audit paths may help, but colour is not part of a codeword. A grayscale image must still reproduce the codebook from bits. Repair missing or duplicated edge bits rather than renaming symbols to evade an error.

Decode the nine-bit message by returning to root

Use the original nine-bit string 010110111. Read 0 to A and return to root; then 10 to B, return; 110 to C, return; finally 111 to D, giving ABCD. The explanatory grouping is 0, 10, 110, 111; commas are not transmitted bits. A remaining 11 stops at internal J and is incomplete, not grounds to guess C or D. Walk the tree manually in notes if no decoder exists; do not invent an in-app success message.

Keep a conflicting codebook as a separate control

Use a separate bad table A=0, B=01, C=10, D=11. A’s 0 prefixes B’s 01, and 010 has two explicit readings: 0|10 gives AC, while 01|0 gives BA. Keep codebooks separate rather than recolouring and mixing them. This demonstrates ambiguity for this string and table, not a claim that every message in every non-prefix code must have multiple readings. Label both groupings and codebook status; the control is not a second equally valid instant-decoding rule. Root-to-leaf edge counts are one for A, two for B and three for C/D, matching lengths 1,2,3,3. Vertical spacing may change for layout, but adding an unlabelled node to align heights breaks edge-count/code-length correspondence. Short-code leaves need not share a horizontal row with long-code leaves; alignment must not change tree structure.

Check unjustified optimal-compression claims

This code uses nine bits for ABCD. With four equally frequent symbols, a fixed two-bit code uses eight for the same four-symbol sequence, so the invented variable code is not automatically smaller. Suitability depends on conditions such as symbol frequencies; no private-file compression test is performed. Inspect exported bits and symbols. If font substitution makes 1 resemble a divider, repair typography or legend rather than the codebook. Hide auxiliary links that could look traversable, retaining mathematical and layout evidence separately.

Reopen and audit leaf paths and reset rules

Save, leave and reopen. Recover each root-to-leaf path, count four leaves/three internals/six edges, then decode nine bits without consulting the old answer. Output at leaves and reset; stopping internally remains incomplete regardless of geometric spacing. Open final delivery and inspect bit labels, bad-table ambiguity and the equal-frequency eight-bit comparison. Leave untested formats unverified. Another reader should reconstruct the codebook, ABCD and both bad-table groupings, not infer in-app compression, decompression or optimality for arbitrary frequencies.