YES

Problem 1: 

(VAR v_NonEmpty:S f:S x:S xs:S)
(RULES
app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
app(app(map,f:S),nil) -> nil
app(app(treemap,f:S),app(app(node,x:S),xs:S)) -> app(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(treemap,f:S),app(app(node,x:S),xs:S)) -> app(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(f:S,x:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(app(map,app(treemap,f:S)),xs:S)
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(node,app(f:S,x:S))
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(f:S,x:S)
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(treemap,f:S),app(app(node,x:S),xs:S)) -> app(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(f:S,x:S))
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(app(map,app(treemap,f:S)),xs:S)
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(node,app(f:S,x:S))
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(f:S,x:S)
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(treemap,f:S),app(app(node,x:S),xs:S)) -> app(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(app(map,app(treemap,f:S)),xs:S)
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(f:S,x:S)
->->-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(treemap,f:S),app(app(node,x:S),xs:S)) -> app(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))

Problem 1: 

Subterm Processor:
-> Pairs:
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(app(map,f:S),xs:S)
 APP(app(map,f:S),app(app(cons,x:S),xs:S)) -> APP(f:S,x:S)
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(app(map,app(treemap,f:S)),xs:S)
 APP(app(treemap,f:S),app(app(node,x:S),xs:S)) -> APP(f:S,x:S)
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(treemap,f:S),app(app(node,x:S),xs:S)) -> app(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))
->Projection:
 pi(APP) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(map,f:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(f:S,x:S)),app(app(map,f:S),xs:S))
 app(app(map,f:S),nil) -> nil
 app(app(treemap,f:S),app(app(node,x:S),xs:S)) -> app(app(node,app(f:S,x:S)),app(app(map,app(treemap,f:S)),xs:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.