Question 35437: Degree-One Leaves of a Finite Tree

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0069-35437 · q-20261007-advance-20261009-r02-tree-0062

Original question and givens

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

G001 — The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

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

Step-by-step derivation

S001 · implies

Before

source roles = [qf-tree-leaf-set, [E, V, domain, operation]]

After

derived = [tree_leaf_set, finite_simple_undirected]

Conditions: none

Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.

Step identifiers and dependencies

Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001

S002 · implies

Before

source roles = [TREE vertices ORIGINAL, 10, [0, 1, 2, 3, 4, 5]]

After

derived = [[0, 1, 2, 3, 4, 5]]

Conditions: none

Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.

Step identifiers and dependencies

Rule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S001

S003 · implies

Before

source roles = [TREE raw edges ORIGINAL, 0, [3, 2]]

After

derived = [[3, 2]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependencies

Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S002

S004 · implies

Before

source roles = [TREE raw edges ORIGINAL, 1, [2, 1]]

After

derived = [[2, 1]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependencies

Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S003

S005 · implies

Before

source roles = [TREE raw edges ORIGINAL, 2, [1, 0]]

After

derived = [[1, 0]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependencies

Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S004

S006 · implies

Before

source roles = [TREE raw edges ORIGINAL, 3, [4, 0]]

After

derived = [[4, 0]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependencies

Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S005

S007 · implies

Before

source roles = [TREE raw edges ORIGINAL, 4, [0, 5]]

After

derived = [[0, 5]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependencies

Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S006

S008 · implies

Before

source roles = [[0, 1, 2, 3, 4, 5], [[0, 5], [1, 0], [2, 1], [3, 2], [4, 0]], Yes]

After

derived = [[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], [[0, [1, 4, 5]], [1, [0, 2]], [2, [1, 3]], [3, [2]], [4, [0]], [5, [0]]]]

Conditions: none

Type the WHOLE raw edge list,then normalize undirected endpoints/deduplicate;reversed duplicates are allowed.INPUT edges have no loops/outside endpoints,while isolated vertices stay in V.TREE raw E does NOT acquire LDL’s200-entry limit.

Step identifiers and dependencies

Rule: Graph_WHOLE_simple_undirected_edges_normalized_NO_loops_or_outside · Uses: G001, S007

S009 · implies

Before

source roles = [0, 1, 0, 1]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependencies

Rule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S008

S010 · implies

Before

source roles = [0, 4, 1, 4]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependencies

Rule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S009

S011 · implies

Before

source roles = [0, 5, 4, 5]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependencies

Rule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S010

S012 · implies

Before

source roles = [1, 2, 5, 2]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependencies

Rule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S011

S013 · implies

Before

source roles = [2, 3, 2, 3]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependencies

Rule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S012

S014 · implies

Before

source roles = [0, [1, 4, 5]]

After

derived = [[0, 1, 4, 5], [0]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependencies

Rule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S013

S015 · implies

Before

source roles = [1, [0, 2]]

After

derived = [[0, 1, 4, 5, 2], [0, 1]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependencies

Rule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S014

S016 · implies

Before

source roles = [4, [0]]

After

derived = [[0, 1, 4, 5, 2], [0, 1, 4]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependencies

Rule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S015

S017 · implies

Before

source roles = [5, [0]]

After

derived = [[0, 1, 4, 5, 2], [0, 1, 4, 5]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependencies

Rule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S016

S018 · implies

Before

source roles = [2, [1, 3]]

After

derived = [[0, 1, 4, 5, 2, 3], [0, 1, 2, 4, 5]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependencies

Rule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S017

S019 · implies

Before

source roles = [3, [2]]

After

derived = [[0, 1, 4, 5, 2, 3], [0, 1, 2, 3, 4, 5]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependencies

Rule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S018

S020 · implies

Before

source roles = [[0, 1, 2, 3, 4, 5], [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]]

After

derived = [Yes, Yes, Yes, Yes]

Conditions: none

THIS entry defines a tree as NONEMPTY,connected and acyclic.Empty graph is No;an edgeless singleton is Yes.A triangle plus an isolated vertex is No EVEN when|E|=|V|−1.

Step identifiers and dependencies

Rule: Tree_NONEMPTY_connected_acyclic_ALL_three · Uses: G001, S019

S021 · implies

Before

source roles = [0, [1, 4, 5]]

After

derived = [3, No]

Conditions: none

THIS entry defines a leaf by degree EXACTLY1.Check EVERY vertex and return a sorted LIST.A singleton has degree0,so its leaf list is[];do not silently use degree≤1.

Step identifiers and dependencies

Rule: Tree_LEAF_EACH_degree_EXACTLY1_singleton0_not_leaf · Uses: G001, S020

S022 · implies

Before

source roles = [1, [0, 2]]

After

derived = [2, No]

Conditions: none

THIS entry defines a leaf by degree EXACTLY1.Check EVERY vertex and return a sorted LIST.A singleton has degree0,so its leaf list is[];do not silently use degree≤1.

Step identifiers and dependencies

Rule: Tree_LEAF_EACH_degree_EXACTLY1_singleton0_not_leaf · Uses: G001, S021

S023 · implies

Before

source roles = [2, [1, 3]]

After

derived = [2, No]

Conditions: none

THIS entry defines a leaf by degree EXACTLY1.Check EVERY vertex and return a sorted LIST.A singleton has degree0,so its leaf list is[];do not silently use degree≤1.

Step identifiers and dependencies

Rule: Tree_LEAF_EACH_degree_EXACTLY1_singleton0_not_leaf · Uses: G001, S022

S024 · implies

Before

source roles = [3, [2]]

After

derived = [1, Yes]

Conditions: none

THIS entry defines a leaf by degree EXACTLY1.Check EVERY vertex and return a sorted LIST.A singleton has degree0,so its leaf list is[];do not silently use degree≤1.

Step identifiers and dependencies

Rule: Tree_LEAF_EACH_degree_EXACTLY1_singleton0_not_leaf · Uses: G001, S023

S025 · implies

Before

source roles = [4, [0]]

After

derived = [1, Yes]

Conditions: none

THIS entry defines a leaf by degree EXACTLY1.Check EVERY vertex and return a sorted LIST.A singleton has degree0,so its leaf list is[];do not silently use degree≤1.

Step identifiers and dependencies

Rule: Tree_LEAF_EACH_degree_EXACTLY1_singleton0_not_leaf · Uses: G001, S024

S026 · implies

Before

source roles = [5, [0]]

After

derived = [1, Yes]

Conditions: none

THIS entry defines a leaf by degree EXACTLY1.Check EVERY vertex and return a sorted LIST.A singleton has degree0,so its leaf list is[];do not silently use degree≤1.

Step identifiers and dependencies

Rule: Tree_LEAF_EACH_degree_EXACTLY1_singleton0_not_leaf · Uses: G001, S025

Reverse checks

S026 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S025 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S024 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S023 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S022 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S021 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S020 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S019 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S018 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S017 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S016 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S015 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S014 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S013 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S012 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S011 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S010 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S009 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S008 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S007 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S006 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S005 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S004 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S003 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S002 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
S001 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.

Result

[3, 4, 5]

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: 7572a94ec4449977d729d2d986ac46fb8680a087fbba46ecc966c0030e2bce7a
candidate record SHA-256: 8a01914f8332ab231ca456d1a263424e30e312a933d9ccb967ae0ce46e9fc08c
semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
Lean: NOT_RUN · human review: NOT_PERFORMED

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