YES

Problem 1: 

(VAR v_NonEmpty:S @l1:S @l2:S @t:S @t1:S @t2:S @x:S @xs:S)
(RULES
append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
append#1(nil,@l2:S) -> @l2:S
subtrees(@t:S) -> subtrees#1(@t:S)
subtrees#1(leaf) -> nil
subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APPEND(@l1:S,@l2:S) -> APPEND#1(@l1:S,@l2:S)
 APPEND#1(::(@x:S,@xs:S),@l2:S) -> APPEND(@xs:S,@l2:S)
 SUBTREES(@t:S) -> SUBTREES#1(@t:S)
 SUBTREES#1(node(@x:S,@t1:S,@t2:S)) -> SUBTREES(@t1:S)
 SUBTREES#1(node(@x:S,@t1:S,@t2:S)) -> SUBTREES#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 SUBTREES#2(@l1:S,@t1:S,@t2:S,@x:S) -> SUBTREES(@t2:S)
 SUBTREES#2(@l1:S,@t1:S,@t2:S,@x:S) -> SUBTREES#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 SUBTREES#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> APPEND(@l1:S,@l2:S)
-> Rules:
 append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
 append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
 append#1(nil,@l2:S) -> @l2:S
 subtrees(@t:S) -> subtrees#1(@t:S)
 subtrees#1(leaf) -> nil
 subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))

Problem 1: 

SCC Processor:
-> Pairs:
 APPEND(@l1:S,@l2:S) -> APPEND#1(@l1:S,@l2:S)
 APPEND#1(::(@x:S,@xs:S),@l2:S) -> APPEND(@xs:S,@l2:S)
 SUBTREES(@t:S) -> SUBTREES#1(@t:S)
 SUBTREES#1(node(@x:S,@t1:S,@t2:S)) -> SUBTREES(@t1:S)
 SUBTREES#1(node(@x:S,@t1:S,@t2:S)) -> SUBTREES#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 SUBTREES#2(@l1:S,@t1:S,@t2:S,@x:S) -> SUBTREES(@t2:S)
 SUBTREES#2(@l1:S,@t1:S,@t2:S,@x:S) -> SUBTREES#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 SUBTREES#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> APPEND(@l1:S,@l2:S)
-> Rules:
 append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
 append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
 append#1(nil,@l2:S) -> @l2:S
 subtrees(@t:S) -> subtrees#1(@t:S)
 subtrees#1(leaf) -> nil
 subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APPEND(@l1:S,@l2:S) -> APPEND#1(@l1:S,@l2:S)
 APPEND#1(::(@x:S,@xs:S),@l2:S) -> APPEND(@xs:S,@l2:S)
->->-> Rules:
 append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
 append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
 append#1(nil,@l2:S) -> @l2:S
 subtrees(@t:S) -> subtrees#1(@t:S)
 subtrees#1(leaf) -> nil
 subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))
->->Cycle:
->->-> Pairs:
 SUBTREES(@t:S) -> SUBTREES#1(@t:S)
 SUBTREES#1(node(@x:S,@t1:S,@t2:S)) -> SUBTREES(@t1:S)
 SUBTREES#1(node(@x:S,@t1:S,@t2:S)) -> SUBTREES#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 SUBTREES#2(@l1:S,@t1:S,@t2:S,@x:S) -> SUBTREES(@t2:S)
->->-> Rules:
 append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
 append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
 append#1(nil,@l2:S) -> @l2:S
 subtrees(@t:S) -> subtrees#1(@t:S)
 subtrees#1(leaf) -> nil
 subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 APPEND(@l1:S,@l2:S) -> APPEND#1(@l1:S,@l2:S)
 APPEND#1(::(@x:S,@xs:S),@l2:S) -> APPEND(@xs:S,@l2:S)
-> Rules:
 append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
 append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
 append#1(nil,@l2:S) -> @l2:S
 subtrees(@t:S) -> subtrees#1(@t:S)
 subtrees#1(leaf) -> nil
 subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))
->Projection:
 pi(APPEND) = 1
 pi(APPEND#1) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 APPEND(@l1:S,@l2:S) -> APPEND#1(@l1:S,@l2:S)
-> Rules:
 append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
 append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
 append#1(nil,@l2:S) -> @l2:S
 subtrees(@t:S) -> subtrees#1(@t:S)
 subtrees#1(leaf) -> nil
 subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 SUBTREES(@t:S) -> SUBTREES#1(@t:S)
 SUBTREES#1(node(@x:S,@t1:S,@t2:S)) -> SUBTREES(@t1:S)
 SUBTREES#1(node(@x:S,@t1:S,@t2:S)) -> SUBTREES#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 SUBTREES#2(@l1:S,@t1:S,@t2:S,@x:S) -> SUBTREES(@t2:S)
-> Rules:
 append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
 append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
 append#1(nil,@l2:S) -> @l2:S
 subtrees(@t:S) -> subtrees#1(@t:S)
 subtrees#1(leaf) -> nil
 subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))
->Projection:
 pi(SUBTREES) = 1
 pi(SUBTREES#1) = 1
 pi(SUBTREES#2) = 3

Problem 1.2: 

SCC Processor:
-> Pairs:
 SUBTREES(@t:S) -> SUBTREES#1(@t:S)
 SUBTREES#2(@l1:S,@t1:S,@t2:S,@x:S) -> SUBTREES(@t2:S)
-> Rules:
 append(@l1:S,@l2:S) -> append#1(@l1:S,@l2:S)
 append#1(::(@x:S,@xs:S),@l2:S) -> ::(@x:S,append(@xs:S,@l2:S))
 append#1(nil,@l2:S) -> @l2:S
 subtrees(@t:S) -> subtrees#1(@t:S)
 subtrees#1(leaf) -> nil
 subtrees#1(node(@x:S,@t1:S,@t2:S)) -> subtrees#2(subtrees(@t1:S),@t1:S,@t2:S,@x:S)
 subtrees#2(@l1:S,@t1:S,@t2:S,@x:S) -> subtrees#3(subtrees(@t2:S),@l1:S,@t1:S,@t2:S,@x:S)
 subtrees#3(@l2:S,@l1:S,@t1:S,@t2:S,@x:S) -> ::(node(@x:S,@t1:S,@t2:S),append(@l1:S,@l2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.