Greetings Karen,
Is river birch susceptible to Verticillium wilt? The answer is, "no". Is eastern redbud and plum susceptible to Verticillium wilt? The answer is, "yes".
Fungicides to treat Verticillium wilt are not recommended or warranted, nor will they cure the fungal infection!!! In order to correctly verify whether or not your plum and redbud have Verticillium wilt, you should submit several samples to Purdue University plant clinic http://www.ppdl.purdue.edu/PPDL/index.html
More info on Verticillium Wilt: http://web.aces.uiuc.edu/vista/pdf_pubs/VERTWILT.PDF