# Question 8451: Degree-One Leaf Deletion without Renumbering [Return to catalogue page 43](https://ocme.evemisslab.com/en/questions/page/43/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r06-tree-foundations/#q-20261012-advance-20261017-r06-tree-0051) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0015-08451 · q-20261012-advance-20261017-r06-tree-0051 ## Original question and givens Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated). - G001 — Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-tree](https://ocme.evemisslab.com/en/mko/mko-tree/) ## Step-by-step derivation S001 · impliesBefore n=0;raw edges=[] After canonical edges=[] Conditions: none Validate every simple undirected edge for typed distinct endpoints.Sort endpoints and count reversed or repeated edges once;do not infer adjacency from a drawing. Step identifiers and dependenciesRule: simple_graph_edge_normalization · Uses: G001 S002 · impliesBefore phase=original;spanning forest=[];all edges=[] After nonforest edges=[];cyclic=No Conditions: none First-discovery edges form a tree in each reached component.Any remaining edge closes an existing tree path into a cycle.Components were established first;this is not an edge-count-only tree test. Step identifiers and dependenciesRule: spanning_forest_cycle_witness · Uses: G001, S001 S003 · impliesBefore phase=original;vertices=[];components=[];cyclic=No After pairwise reachable=Yes;nonempty=No;tree=No Conditions: none A tree must be nonempty,have one component and be acyclic.Pairwise reachability is vacuously true on an empty vertex set,but nonemptiness fails.A singleton with no edges is a tree;its degree0 vertex is not a leaf. Step identifiers and dependenciesRule: nonempty_connected_acyclic_tree_gate · Uses: G001, S002 S004 · impliesBefore original tree=No;at=0;typed=No;degree/status=invalid vertex After qualified degree-one leaf=No Conditions: none Require an original tree,a typed vertex and original degree exactly1.The degree0 vertex of a singleton tree is not a leaf. Step identifiers and dependenciesRule: original_tree_degree_one_leaf_guard · Uses: G001, S003 ## Reverse checks - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (0, [], No, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 844492762340fe8c8aeff766a2313785daf0cdf966722092463cb48876772e8e - candidate record SHA-256: cb146d65946aa39edaae89a5e0c264fa50d9b626d47db43671fda61aeaa15691 - semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18 - question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r06-tree-0051.json)