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I'm not sure what type they are, but it is similar to the MacArthur palm (Ptychosperma macarthurii) - the similarties are that it grows in clusters, and has a similar seed thing, but I never remember seeing red seeds on it. I'd say it's definitely some kind of Ptychosperma, but which one I'm not sure...
A few days ago, I noticed that the seed/flower stalks had burst into bloom - I'd never noticed the flowers there before. Maybe it's the recent hot and dry weather.
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Closer up, I saw that it was buzzing with bees!
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On this one you can see the pollen that has been collected (a 'pollen sac'?)
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Based on the picture of the flowers
In fact, I have taken to pulling off the flower stalk, as my father-in-law suggested - in order to stop lots of little trees growing up. But now I feel kind of guilty, having seen the bees enjoying the flowers so much!
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It reminded me of one of the paradoxes of gardening - I like to think of myself as using the garden to get closer to nature, using organic means, caring for the environment and all that; but in the end of the day, a garden is such an unnatural environment :/