Why are the branches of my
contorted filbert dying?

Contorted filbert, also called Harry Lauder's Walking Stick is susceptible to a blight which is what TSU Extension Agent Joellen Dimond thinks is going on here. Joellen points out that this year there has been a lot of rain which creates the conditions the blight likes. There is not much you can do except try to adjust the environmental conditions. Make sure the plant is not is a low spot and there is good drainage and good air circulation around the leaves. If you do need to replace it, don’t plant it in the same spot. You can purchase a contorted filbert from most nurseries. ​

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Blight
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Related Resources:
Picture of the Week: Contorted Filbert Blight
Plant of the Week Harry Lauder's Walking Stick