2b:null 2c:["$","title",null,{"children":"Tangled tree visualization | brettforbes | Observable"}] 31:T510,intro = md`A tree with multiple inheritance (sometimes called *tangled tree*) cannot be represented by using a classic tree visualization. It is technically a directed acyclic graph (DAG) with one (or more) nodes identified as root. Using a graph visualization technique solves the issue, but poorly represents some peculiarities of a quasi-tree structure. This visualization tries to address the representation of a tangled tree (a genealogy in this specific example). The layout and basic technique is the same underlying [Geneaquilts](https://aviz.fr/Research/Geneaquilts) [[1](#footnote1)], but with curved links instead of matrices. Metro-style bundling is used when a parent node appears in multiple "families", and the layout is made more compact on the y axis. The intention is for the graphic to be more readily understood, while losing some of the advantages of Geneaquilts (e.g., representing a union without children). See also [this older, simpler example](https://beta.observablehq.com/@nitaku/tangled-tree-visualization). The technique can also be applied to other types of tangled trees. Such structures are commonly found in computational linguistics (thesauri, some ontologies, wordnets). (Thanks to [@trebor](https://observablehq.com/@trebor) for making it more reusable!)`32:T668,renderChart = (data, options={}) => { options.color ||= (d, i) => color(i) const tangleLayout = constructTangleLayout(_.cloneDeep(data), options); return svg``; }33:Tb89,data = [ [{ id: 'Chaos' }], [{ id: 'Gaea', parents: ['Chaos'] }, { id: 'Uranus' }], [ { id: 'Oceanus', parents: ['Gaea', 'Uranus'] }, { id: 'Thethys', parents: ['Gaea', 'Uranus'] }, { id: 'Pontus' }, { id: 'Rhea', parents: ['Gaea', 'Uranus'] }, { id: 'Cronus', parents: ['Gaea', 'Uranus'] }, { id: 'Coeus', parents: ['Gaea', 'Uranus'] }, { id: 'Phoebe', parents: ['Gaea', 'Uranus'] }, { id: 'Crius', parents: ['Gaea', 'Uranus'] }, { id: 'Hyperion', parents: ['Gaea', 'Uranus'] }, { id: 'Iapetus', parents: ['Gaea', 'Uranus'] }, { id: 'Thea', parents: ['Gaea', 'Uranus'] }, { id: 'Themis', parents: ['Gaea', 'Uranus'] }, { id: 'Mnemosyne', parents: ['Gaea', 'Uranus'] } ], [ { id: 'Doris', parents: ['Oceanus', 'Thethys'] }, { id: 'Neures', parents: ['Pontus', 'Gaea'] }, { id: 'Dionne' }, { id: 'Demeter', parents: ['Rhea', 'Cronus'] }, { id: 'Hades', parents: ['Rhea', 'Cronus'] }, { id: 'Hera', parents: ['Rhea', 'Cronus'] }, { id: 'Alcmene' }, { id: 'Zeus', parents: ['Rhea', 'Cronus'] }, { id: 'Eris' }, { id: 'Leto', parents: ['Coeus', 'Phoebe'] }, { id: 'Amphitrite' }, { id: 'Medusa' }, { id: 'Poseidon', parents: ['Rhea', 'Cronus'] }, { id: 'Hestia', parents: ['Rhea', 'Cronus'] } ], [ { id: 'Thetis', parents: ['Doris', 'Neures'] }, { id: 'Peleus' }, { id: 'Anchises' }, { id: 'Adonis' }, { id: 'Aphrodite', parents: ['Zeus', 'Dionne'] }, { id: 'Persephone', parents: ['Zeus', 'Demeter'] }, { id: 'Ares', parents: ['Zeus', 'Hera'] }, { id: 'Hephaestus', parents: ['Zeus', 'Hera'] }, { id: 'Hebe', parents: ['Zeus', 'Hera'] }, { id: 'Hercules', parents: ['Zeus', 'Alcmene'] }, { id: 'Megara' }, { id: 'Deianira' }, { id: 'Eileithya', parents: ['Zeus', 'Hera'] }, { id: 'Ate', parents: ['Zeus', 'Eris'] }, { id: 'Leda' }, { id: 'Athena', parents: ['Zeus'] }, { id: 'Apollo', parents: ['Zeus', 'Leto'] }, { id: 'Artemis', parents: ['Zeus', 'Leto'] }, { id: 'Triton', parents: ['Poseidon', 'Amphitrite'] }, { id: 'Pegasus', parents: ['Poseidon', 'Medusa'] }, { id: 'Orion', parents: ['Poseidon'] }, { id: 'Polyphemus', parents: ['Poseidon'] } ], [ { id: 'Deidamia' }, { id: 'Achilles', parents: ['Peleus', 'Thetis'] }, { id: 'Creusa' }, { id: 'Aeneas', parents: ['Anchises', 'Aphrodite'] }, { id: 'Lavinia' }, { id: 'Eros', parents: ['Hephaestus', 'Aphrodite'] }, { id: 'Helen', parents: ['Leda', 'Zeus'] }, { id: 'Menelaus' }, { id: 'Polydueces', parents: ['Leda', 'Zeus'] } ], [ { id: 'Andromache' }, { id: 'Neoptolemus', parents: ['Deidamia', 'Achilles'] }, { id: 'Aeneas(2)', parents: ['Creusa', 'Aeneas'] }, { id: 'Pompilius', parents: ['Creusa', 'Aeneas'] }, { id: 'Iulus', parents: ['Lavinia', 'Aeneas'] }, { id: 'Hermione', parents: ['Helen', 'Menelaus'] } ] ]34:T1089,constructTangleLayout = (levels, options={}) => { // precompute level depth levels.forEach((l, i) => l.forEach(n => (n.level = i))); var nodes = levels.reduce((a, x) => a.concat(x), []); var nodes_index = {}; nodes.forEach(d => (nodes_index[d.id] = d)); // objectification nodes.forEach(d => { d.parents = (d.parents === undefined ? [] : d.parents).map( p => nodes_index[p] ); }); // precompute bundles levels.forEach((l, i) => { var index = {}; l.forEach(n => { if (n.parents.length == 0) { return; } var id = n.parents .map(d => d.id) .sort() .join('-X-'); if (id in index) { index[id].parents = index[id].parents.concat(n.parents); } else { index[id] = { id: id, parents: n.parents.slice(), level: i, span: i - d3.min(n.parents, p => p.level) }; } n.bundle = index[id]; }); l.bundles = Object.keys(index).map(k => index[k]); l.bundles.forEach((b, i) => (b.i = i)); }); var links = []; nodes.forEach(d => { d.parents.forEach(p => links.push({ source: d, bundle: d.bundle, target: p }) ); }); var bundles = levels.reduce((a, x) => a.concat(x.bundles), []); // reverse pointer from parent to bundles bundles.forEach(b => b.parents.forEach(p => { if (p.bundles_index === undefined) { p.bundles_index = {}; } if (!(b.id in p.bundles_index)) { p.bundles_index[b.id] = []; } p.bundles_index[b.id].push(b); }) ); nodes.forEach(n => { if (n.bundles_index !== undefined) { n.bundles = Object.keys(n.bundles_index).map(k => n.bundles_index[k]); } else { n.bundles_index = {}; n.bundles = []; } n.bundles.sort((a,b) => d3.descending(d3.max(a, d => d.span), d3.max(b, d => d.span))) n.bundles.forEach((b, i) => (b.i = i)); }); links.forEach(l => { if (l.bundle.links === undefined) { l.bundle.links = []; } l.bundle.links.push(l); }); // layout const padding = 8; const node_height = 22; const node_width = 70; const bundle_width = 14; const level_y_padding = 16; const metro_d = 4; const min_family_height = 22; options.c ||= 16; const c = options.c; options.bigc ||= node_width+c; nodes.forEach( n => (n.height = (Math.max(1, n.bundles.length) - 1) * metro_d) ); var x_offset = padding; var y_offset = padding; levels.forEach(l => { x_offset += l.bundles.length * bundle_width; y_offset += level_y_padding; l.forEach((n, i) => { n.x = n.level * node_width + x_offset; n.y = node_height + y_offset + n.height / 2; y_offset += node_height + n.height; }); }); var i = 0; levels.forEach(l => { l.bundles.forEach(b => { b.x = d3.max(b.parents, d => d.x) + node_width + (l.bundles.length - 1 - b.i) * bundle_width; b.y = i * node_height; }); i += l.length; }); links.forEach(l => { l.xt = l.target.x; l.yt = l.target.y + l.target.bundles_index[l.bundle.id].i * metro_d - (l.target.bundles.length * metro_d) / 2 + metro_d / 2; l.xb = l.bundle.x; l.yb = l.bundle.y; l.xs = l.source.x; l.ys = l.source.y; }); // compress vertical space var y_negative_offset = 0; levels.forEach(l => { y_negative_offset += -min_family_height + d3.min(l.bundles, b => d3.min(b.links, link => link.ys - 2*c - (link.yt + c)) ) || 0; l.forEach(n => (n.y -= y_negative_offset)); }); // very ugly, I know links.forEach(l => { l.yt = l.target.y + l.target.bundles_index[l.bundle.id].i * metro_d - (l.target.bundles.length * metro_d) / 2 + metro_d / 2; l.ys = l.source.y; l.c1 = l.source.level - l.target.level > 1 ? Math.min(options.bigc, l.xb-l.xt, l.yb-l.yt)-c : c; l.c2 = c; }); var layout = { width: d3.max(nodes, n => n.x) + node_width + 2 * padding, height: d3.max(nodes, n => n.y) + node_height / 2 + 2 * padding, node_height, node_width, bundle_width, level_y_padding, metro_d }; return { levels, nodes, nodes_index, links, bundles, layout }; }35:[] 2f:{"id":"921667bef36758e7","slug":null,"title":"Tangled tree visualization","thumbnail":"d83a81ce0f4497e36616521f5b5a58c34f90aaacd515bc861ed15292e554f10f","access_level":"public","create_time":"$D2026-06-12T11:05:24.758Z","update_time":"$D2026-06-12T11:05:24.758Z","view_time":null,"first_public_time":"$D2026-06-12T11:05:24.758Z","has_unpublished_edits":null,"fork":{"id":"4cab522856cbfdde","slug":"tangled-tree-visualization-ii","title":"Tangled tree visualization","owner":{"id":"26c2ccc018e1ea02","login":"nitaku","name":"Matteo Abrate"}},"stars":0,"starred":false,"owner":{"type":"team","id":"f846f459b144e893","login":"brettforbes","name":"","avatar_url":"https://avatars-next.observableusercontent.com/90abd4233f342d3bde898a6f28cf0f1a64652b13845939fedb9e12eda77f7bd0","tier":"starter_2024"},"version":670,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"e3ae8b6a8e4f693b","login":"brettforbes","name":"","avatar_url":"https://avatars-next.observableusercontent.com/90abd4233f342d3bde898a6f28cf0f1a64652b13845939fedb9e12eda77f7bd0","tier":null}],"members":[{"type":"individual","id":"e3ae8b6a8e4f693b","login":"brettforbes","name":"","avatar_url":"https://avatars-next.observableusercontent.com/90abd4233f342d3bde898a6f28cf0f1a64652b13845939fedb9e12eda77f7bd0","tier":null,"role":"editor"},{"type":"team","id":"f846f459b144e893","login":"brettforbes","name":"","avatar_url":"https://avatars-next.observableusercontent.com/90abd4233f342d3bde898a6f28cf0f1a64652b13845939fedb9e12eda77f7bd0","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Tangled tree visualization","cells":[{"id":0,"value":"md`# Tangled tree visualization`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":527,"value":"renderChart(data)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":226,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":556,"value":"md`## Code`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":523,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":239,"value":"color = d3.scaleOrdinal(d3.schemeDark2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":435,"value":"background_color = 'white'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":532,"value":"md`## Dependencies`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":242,"value":"d3 = require('d3-scale', 'd3-scale-chromatic', 'd3-array')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":521,"value":"_ = require(\"lodash\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":546,"value":"import {footnote} from \"@jashkenas/markdown-style-footnotes\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":535,"value":"md`## References`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":539,"value":"footnote1 = footnote(\"intro\", 1, `A. Bezerianos, P. Dragicevic, J. Fekete, J. Bae and B. Watson, \"GeneaQuilts: A System for Exploring Large Genealogies,\" in IEEE Transactions on Visualization and Computer Graphics, vol. 16, no. 6, pp. 1073-1081, Nov.-Dec. 2010, doi: 10.1109/TVCG.2010.159.`)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 30:"$2f:body"