Our efforts having been in vain, we had abandoned the matter, hoping that it might turn up of itself one day. 
( ( ( Our efforts ) ( having ( been ( in vain ) ) ) ) ( , ( we ( ( had ( ( ( abandoned ( the matter ) ) , ) ( hoping ( that ( it ( might ( ( ( turn up ) ( of itself ) ) ( one day ) ) ) ) ) ) ) ) . ) ) ) )
(ROOT (S (S (NP (PRP$ Our) (NNS efforts)) (VP (VBG having) (VP (VBN been) (PP (IN in) (NP (JJ vain)))))) (, ,) (NP (PRP we)) (VP (VBD had) (VP (VBN abandoned) (NP (DT the) (NN matter)) (, ,) (S (VP (VBG hoping) (SBAR (IN that) (S (NP (PRP it)) (VP (MD might) (VP (VB turn) (PRT (RP up)) (PP (IN of) (NP (PRP itself))) (NP (CD one) (NN day)))))))))) (. .)))
Even though we had not solved the problem, we kept on trying.
( ( Even ( though ( we ( ( had not ) ( solved ( the problem ) ) ) ) ) ) ( , ( we ( ( kept ( on trying ) ) . ) ) ) )
(ROOT (S (SBAR (RB Even) (IN though) (S (NP (PRP we)) (VP (VBD had) (RB not) (VP (VBN solved) (NP (DT the) (NN problem)))))) (, ,) (NP (PRP we)) (VP (VBD kept) (PP (IN on) (S (VP (VBG trying))))) (. .)))